The class {dog} is stipulated to be a subset of the class {animal}. The other details about {dogs} and {animals} are referenced in the axiomatic model of the actual world knowledge ontology inheritance hierarchy. — PL Olcott

Isn't axiomatic model for formalizing various branches of mathematical theory, including geometry, algebra, set theory? Applying that concept to linguistic topic sounds incorrect.

What is the "whole confusion"? Yes, there are people who don't know about set theory and are confused about it so that they make false and/or confused claims about it. But the axioms of set theory don't engender a confusion. They engender philosophical discussion and debate, but there is no confusion as to what is or is not proven in set theory. Whether any given axiom is wrong or not is a fair question, but it doesn't justify people who don't know anything about axiomatic set theory thereby spreading disinformation and their own confusions about it. — TonesInDeepFreeze

What I meant was that, as Frege, Russell, Wittgenstein and Hilbert had in their minds, that many math axioms, concepts and definitions are not logical or justifiable in real life truths. A good example is the concept of Infinity, and Infinite Sets.

Infinity is not numeric, but a property of motions, operations and actions. But they seem to think it is some solid existence in reality. When they talk about the concepts like infinite sets and claim this or that as if there are self-evident truths for them, it sounds confused.

Of course. And I have many times explicitly said that no one is obligated to accept, like, or work with any given set of axioms and inference rules. But if the axioms and inference rules are recursive, no matter what else they are, then it is objective to check whether a given sequence purported to be a proof sequence is indeed a proof sequence per the cited axioms and rules. If you give me formal (recursive) axioms and rules of your own, and a proof sequence with them, then no matter whether I like your axioms or rules, I would confirm that your proof is indeed a proof from those axioms and rules. — TonesInDeepFreeze

In Philosophy, they don't use axioms and deductive reasonings and proofs as their main methodology.  Philosophy can check the axioms, theorems, hypotheses, definitions and even the questions statements for their validity, but the actual proof processes and math knowledge themselves are not the main philosophical interests.

As a "set" is an object it can have a size, and therefore there can be different sizes of sets.

However, as the qualifier "that can be added to" is not an aspect of the size of the set, whilst the expression "different sizes of sets" is grammatical, the expression "different sizes of infinite sets" is ungrammatical. — RussellA

The whole confusion resulted from the wrong premise that infinite numbers do exist. No they don't exist at all. So it is an illusion. From the illusive premises you can draw any conclusions which are also illusive.

Infinity in math has been improvised to explain and describe continuous motion hence the Limit and Integral symbols in Calculus. But they have taken the concept further to apply into the set and number theories. Yes depending on what you accept, you can say the infinite Sets can have different sizes etc. It is OK to keep on saying that in math forums, and it sounds correct because that is what the textbook says.

But when it comes to under the Philosophical analysis, one cannot fail to notice the whole picture was based on the fabricated concepts, which are not very useful or practical in the real world.

One is free to say that we don't need utter objectivity, but then we may say, "Fair enough. So your desideratum is different from those using the axiomatic method." — TonesInDeepFreeze

My statements were from my reasoning. But what you claim to be objectivity is from the textbooks. Please bear in mind, the textbooks are also written by someone who have been reasoning on the subject. It is not the bible, to which you have to take every words and sentences as the objectivity that everyone on the earth must follow. That sounds religious.

Mathematics is a narrow scoped subject which borrowed most of its concepts from Philosophy and modified to suit their abstraction to justify their theorems. Hence we find lots of confusions in math and also the math students. Philosophy can clarify some of its modified concepts for the real meaning of them, so they can understand the subject better.

So you have no reason to believe in the existence of the things behind you? When you put the cup in the cupboard, you cease to have any reason to believe that the cup is in the cupboard?

That's not right. — Banno

There are many other things that can be discussed in the thread such as the world itself, God, Souls, places one never has been, people one never met ... etc. The building which stood across the road, but demolished for the new development, hence no longer existing etc.

There are lots of meat in the tittle of the thread for good classic and traditional philosophising too such as reasons (logical grounds), beliefs (grounded or groundless beliefs) and the existence of the world ... etc.

But you keep pointing out the cup and ask if there is point in the thread sounds some sort of obsession with the cup. That comment sounds very silly. I know the cup exists, but I can start doubting if there is a reason to doubt it does. Why is it silly to doubt whatever it might be, if one has a reason to doubt?

For the same logic, Why is it silly to believe in whatever it might be, if one has a reason to believe? Discussing on the nature of the beliefs and doubts and logical grounds for them is an interesting philosophical topic anyone would say, apart from you.

Only in the sense that facts can be looked up in an encyclopedia and encyclopedias can be updated with new facts. Actual interaction with the world that requires sense input from the sense organs is specifically excluded from the body of analytic knowledge. That dogs exist is analytic. That there is a small black dog in my living room right now is synthetic. — PL Olcott

Analytic sentences are known to be superfluous for the meanings are already in the sentence, and it is just repeating what is in it.

A bachelor is a man who is not married.
A bachelor = a man who is not married
A bachelor is a bachelor, or a man who is not married is a man who is not married.

It also creates some grammar confusions.
No bachelor is married.
No man who is not married is married.
A man who is not married is married.

That dogs exist is analytic. — PL Olcott

That dogs exist is ambiguous. It doesn't say where and when that dogs exist.
It only makes sense if that dogs exist in the real world, and if the sentence has been denoting for the info and also the evidence of the existence.

That dogs exist is analytic is ambiguous in another way that, it sounds like you are claiming that that dogs are analytic. I have not seen or heard analytic dogs. What breed are they? Or do you mean the dogs analyse something? Do the synthetic dogs exist too?

That is an idiomatic reference that does not pertain to the same GUID. — PL Olcott

I think his point is that an analytic system must be able to interact with the external world input data for it to be useful.

This is more clear when we understand that the above finites strings of {"Bachelor", "Male", "Adult", "¬Married"} are totally different across different human languages. — PL Olcott

Yes, but Quine might ask, what about in the case of, when a married woman claims that she is a Bachelor, and you ask how is it possible? She replies "My names is a Bachelor."

Extended real number line — Michael

"What is the number line to infinity?
For instance the number line has arrows at the end to represent this idea of having no bounds. The symbol used to represent infinity is ∞. On the left side of the number line is −∞ and on the right side of the number line is ∞ to describe the boundless behavior of the number line.11 Sept 2021" - Google

It is an entity iff it exists. — Bob Ross

Things exist in minds as well as in empirical world. When things exist in mind, they are called concepts and ideas.

But this is not the argument in this thread. That is specifically about not believing that something continues to exist, unperceived. A very silly argument. — Banno

Many believe in the existence they don't perceive such as God, Souls, afterlife, the places they have never been but seen on the social media and people they have never met but heard of ... etc. How is it silly asking logical ground for the belief? It is silly if and only if you don't understand the question.

I agree. "Infinite" is a property attached to an object, such as "large house" or "infinite number".

As "large" doesn't exist as an object, "infinite" doesn't exist as an object. — RussellA

If that is the case, then it seems barmy to talk about different size of the infinite sets.

I said math and philosophy have different way of doing things
— Corvus

They certainly do, which is why I’m wondering what a thread on mathematics is doing on a philosophy forum. — Joshs

Mathematics must have been believing in Philosophy's assistance in clarifying the tricky concepts. :snicker:

Yes, for example as in "infinite number" where "infinite" is a property of "number". — RussellA

There is no such a thing as "infinite" number. See this is an illusion, and source of the confusion.
Infinity is a property of motion or action, nothing to do with numbers. Infinite number means that you keep adding (or counting whatever) what you have been adding (or counting) to the existing number until halted by break signal (as can be demonstrated in computer programming).

A set containing 3 numbers can be made infinite, when it is in the counting Loop 1, 2, 3, 1, 2, 3 .... ∞ Therefore a term "infinite number" is a misnomer. I bet my bottom dollar that you will never find a number which is infinite, because it doesn't exist. If it did exist, then it is not an infinite number.

Umm... that's a school math book. Have you even studied a math course in the University? They are a bit different. — ssu

I have a few university Calculus and Algebra and Trigonometry books lying around here, and they are full of questions and answers. Studying math means you read the definitions in the books and work on the questions for the answers purely using your reasonings.

And if you study philosophy, you will similarly (hopefully) be given a exam where you have to answer too. — ssu

No. That is not the case. If you study philosophy for the degree, you must read, and write dissertations which you must defend it at a 'viva voce'.

Math and Science pursues the answers in the answer book.
— Corvus
What answer book? — ssu

Have you not read a single math book? If you read any math book, it will have Exercises and Examples after or in the middle of a chapter. The answers for the Exercises will be either at the back of the book, or as a separate Answer Book that you must acquire, if you needed it.

What answer book?

I think mathematics is especially interested in logic. I would dare to say that math is part of logic. — ssu

Sure, not denying that at all. They are all parts of each other we could say that. They are all inter-related too. But the methodologies they employ and the ideas of their goals might be different depending on the folks who are doing them.

Just look at ↪Lionino wrote above. Now I don't know if he is a mathematician, but at least he totally understands that philosophy is part of mathematics. — ssu

Never said math is not part of philosophy. That is what you are saying for some reason.
I said math and philosophy have different way of doing things.

We don't. He proved that they are the same size. — TonesInDeepFreeze

From the point of the set N, it looks like it is. But from the point of the set P, it looks like it is only a half set to N. What's going on?

? :yikes:

I don't get your point here. — ssu

Math and Science pursues the answers in the answer book. You are either right or wrong. Philosophy is more into your arguments and logic for the answers, hence there is no such thing as the answers in the answer book i.e. truth and falsity they pursue are different in nature.

I will likely make a thread about the Grundlagenkrise in the coming weeks. — Lionino

Blimey I was going to make a thread about "Science as a superstition".

I think you got it a bit wrong. Those who are obsessed about truth or falsity are mathematicians. Even if they sometimes have different axiomatic systems, then it's about right or wrong in that formal system. — ssu

I think you got it wrong too. Philosophers don't care about the truths and falsity as the answers in the answer sheets. Philosophers are more concerned with the truth and falsity in the concepts, propositions, and logic.

It's the Philosophers who are interested about a lot more. Things like morals or aesthetics, which obviously aren't about truth or falsity. — ssu

Yes, Philosophy used to be the parents of all sciences and mathematics. It is the mother of all subjects, and we cannot deny the fact.

In Philosophy, they tend to analyse concepts and propositions for truth or falsity. That's what they do. End of the story.

But maybe the mathematicians and scientists do things differently. They don't ask what the concepts mean as long as they are in the textbook. They just accept them, and work on.

"INFINITY definition =1. time or space that has no end: 2. a place that is so far away that it cannot be reached:" - the Cambridge English Dictionary.

It implies that if you know what it is, then you don't know what it is. If you don't know what it is, then you know that you don't know. It is a paradoxical concept, which has to be branded as a contradiction in Philosophy.

Sure, we know that at least a world exists, the world being our mind. But we do not know whether there is an outside world (brain in a vat), that is usually what people talk about when we say the world exists or not. — Lionino

Once you closed eyes and blocked your ears and nose, from the moment, your beliefs and inferences based on your memory of the facts, takes over on the existence of the world outside of you.

Sorry I can't understand, I think this sentence has some words missing. — Lionino

Seeing wave of gravity and saying it is time or space time is like saying, an eclipse is God's facial expression. Just a metaphor or simile whatever you call it. :) Are you a French or Greek?

Here is a finite definition of an infinite set: "A given set S is infinite iff there exists a bijective function between S and a proper subset of S." Furthermore, such a bijective function can be stated finitely.

Here is an example. Take the set of natural numbers ℕ = { 0, 1, ··· }. Now take a proper subset of ℕ containing only even the numbers, ℙ = { 0 , 2 , ··· }. These two are equinumerous because there is a bijective function f : ℕ → ℙ, given by f(n) = 2n.

The proof that "f" is bijective is finite. So is the proof that ℙ is a proper subset of ℕ. — DanCoimbra

Great post, thanks. How do you prove then N is different size to P?

We adduce a sentence G that is is true (to be more precise, it is true in the standard model for the language of arithmetic) if and only if G is not provable in T.

Then we prove that G is not provable in T. So G is a true sentence that is not provable in T. Moreover we show also that ~G is not provable in T. So T is incomplete. — TonesInDeepFreeze

Could you demonstrate and prove the provability and unprovability of G in real arithmetic sentences in T?

Well, you can see gravitational waves insofar as you observe them by checking the spatial distortion that they cause. Maybe that is what they were getting at but I did not see that thread. Not sure what the connection is with what I said though. — Lionino

Maybe they did. But whatever they saw, equating it to time or spacetime sounds bizarre.

For someone who defends physicalism, they are. — Lionino

It would be a form of totemism in disguise for science. Seeing an eclipse, and saying that must a God annoyed at something. A similar logic.

I would say no because those facts could be a fabrication of the mind. — Lionino

The fabrication of the mind is the world. No? I am sure when one dies, his world dies too, because he can no longer fabricate anything anymore.

There cannot be such a thing as a ‘epistemic entity’ because it is, when taken literally, a contradiction in terms: an entity implies something within the ontology of reality, and epistemology pertains solely to knowledge (and specifically not ontology). — Bob Ross

How would you know if something is an entity without knowing what it is?

Ok, so ‘□∀M -> □∃T’ is ‘it is necessary that every motion is ??? and that entails that it is necessary that there exists a time”. That doesn’t make any sense to me. — Bob Ross

That is not propositional logic. It is an EL (Epistemic Logic) operator which means, Agent "knows". It could have been "K" for knows in general, but the box implies knowing via observation.

‘∃M1t1∃M2t2 →□Ag,T,M’ means ‘there exists a motion and time such that there exists another motion and time’ and that entails ‘it is necessary that there is an agento, time, and motion’. Again, I don’t know what this is trying to convey. — Bob Ross

If there was Motion1 to Motion2 with time1 to time2, then the Agent knows Time generated from the Motion via Observation. This is what it means.

One can talk about infinity conceptually, as one does in mathematics, without reference to its empirical verifiability. — DanCoimbra

"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." - Philosophical grammar, p483. Wittgenstein.

Welcome to TPF~ :cool: :up:

How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached? — Philosopher19

Doesn't infinity mean endless? i.e. unreachable eternal continuation in concept?
If it was reachable, then it wouldn't be infinity. Any set or size would be unknowable, if it were infinity. Therefore talking about different size, set or number of infinity, is it not a nonsense?

Sure. But we can see also Quine's point that analytic expressions are meaningless, because the meanings of the words change through time. And most of all, there is no logical explanation why words have the meanings.

For example, the word bachelor's meaning "an unmarried adult male", Quine asks who on earth gave that meaning to bachelor, and why? Meanings of words are totally contingent and changeable. A single words can have also multiple meanings too which adds to the confusion. Hence without the empirical perception which reflects the situation, analytic words themselves have no meanings.

But a physicalist will say that there is only the ink down on the paper, and that any content represented by it exists as chemical reactions in our mind. — Lionino

It seems like some form of superstition. A couple of days ago in one of the new thread here, the OP was claiming that he witnessed the actual wave of gravity with telescope, and it must be the physical existence of spacetime. It sounded like some religious beliefs of some cult folks claiming the earthquakes and hurricanes are act of the angry God or something.

So for physicalists, facts are physical or there are no facts; — Lionino

We are not denying the existence of physicals or substances, but they themselves are not facts or minds.

otherwise it would depend on whether you are talking about the type or the token, or whether the guy you are asking is an idealist, or what the fact is talking about. — Lionino

Wittgenstein said in TLP "The world is the totality of facts.", and it sounds interesting. It also sounds a kind of Solypsism. It cannot be said, but it presents itself. One's perception of the world is limited by one's knowledge of the facts of the world that one knows. The facts includes certain possibilities, impossibilities and logic that operates in the world. Could the facts one knows about the world he faces, and lives in, be the ultimate reason to believe in the existence of the world?

Zen is known for this, for example the book Zen Mind, Beginner’s Mind.
Also, the Tao Te Ching can be read and (somewhat understood) by nearly anyone in an hour.
In mystic Christianity, Jesus’s encouragement to become as children… etc etc. — 0 thru 9

Zen would be a knowledge that is impossible to demonstrate due to the nature of the knowledge, which is subjective and intuition based.

I have no idea what Tao Teching would be. Never heard of it in my puff.

For Jesus and Christianity, I know very little too. Only thing I know is that Jesus has died, but resurrected in 3 days (hence the Easter day). After the resurrection, we don't know where on earth he has been living. This cries for an esoteric inquiry.

I read a little about the underground religious sector stemmed out of Christianity called Gnosis, which is heavily into pagan rituals. These folks would be deeply into esoteric knowledge.

But it seems evident that none of these folks above would agree to demonstrate their esoteric or mystic knowledge even if they knew what they are.

I gave some examples and here is a compiled list:

Brain; (thoughts, thinking) — Mark Nyquist

It looks like a good name for a function in A.I. programming. You could write more details of procedure in the function specifying the variables, constants, inputs and outputs for the different external events fed into the function, and the procedures within the function could go through preset calculations and operations based on the set algorithms from the input and outputs from the hardware sensors in the AI agent.

The functions can be called from various parts of the main program carrying out tasks for the different physical and mental events. When fully implemented, it would be an interesting project or demonstration how AI can simulate the brain functions for generating and interacting with the input and output from the external world, and for all those different mental events and operations.

Cool. What program language are you using for creating the AI system?
Any data structure in hand for the project?

1. G is provable. So G is unprovable
2. G is not provable

So, there is G in the theory T

Have I got it right? — TheMadFool

Shouldn't G be in the form of arithmetic calculus propositions for the incomplete theorem to apply?

But what would be the point of the formalised analytic data, if it is only dictionary nature unable to interact with the external world events and inputs?

In that case it would remain hidden from us because it is beyond our abilities to comprehend it. — Fooloso4

When the mysteries are revealed then they are for the initiated no longer mysteries. — Fooloso4

Sure. My point was that in either case, the knowledge is not for demonstrating to public whether you are able to comprehend it or not. Even if you were a new initiate to the secret society, they will make you to work from the bottom to the top with dedication and hard work for acquiring the knowledge. They won't demonstrate the mystic knowledge, and show you the whole lot at one go, just because you joined their school or club.

https://en.wikipedia.org/wiki/Cyc — PL Olcott

That looks an interesting project. :ok:

No, the observed motion was a change in the fabric of spacetime. — MoK

It sounds like a voice from the deepest well of confusion. Will leave you to it. :yawn: