As I said above, you will double down. You will also seek to obfuscate and change topic. But here, your error is clear. The subject of the quote is not the infinite, but mathematician's discussions of the infinite. — Banno

This part is your usual modus operandi, which is ad hominem and straw man.

My ground involve reading what Wittgenstein says: "mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." He is not saying that infinity is finite, but that the discussions of mathematicians are finite. — Banno

So which discussion is not finite in that case? Does any discussion under the sun go on forever? It doesn't make sense.
Are you possibly suggesting Wittgenstein would have meant that obvious cliche in his writings?

This is your modus operandi. — Banno

Describe "infinity" in clear and actual way in understandable language, and I will tell you about your modus operandi.

No, Tones took up what you said, asking you to justify it. You are in error, both in claiming "Problem with Set Theory is that their concept "infinite" means "finite" and in attributing anything like that to Wittgenstein. — Banno

Yup, that was my interpretation of Wittgenstein. What is your ground for saying it error?

Hmm. You misattributed a position to Wittgenstein. He did not say that "infinite" means "finite". — Banno

"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. — Corvus

Wasn't he saying clearly mathematician's infinite are finite?

What set theory textbook, or any reference in set theory or mathematics, says that 'infinite' means 'finite'? — TonesInDeepFreeze

You misunderstood. It meant that Wittgenstein said that mathematician's infinite means finite in his writings. See the quote above.

Then you said, infinite is not finite, but "not finite". I asked for the textbook definition for infinite in math. Again, my point on it is that, infinity is an abstract concept which has no referent object.

You quoted him, in another thread, as saying — Banno

Too many threads on infinity. You found it OK. Anyway, it wasn't far.
Tone was in the thread, and he would have seen it.

Anyway, Cantor and Dedekind wouldn't have opposed to infinity in set theory, because they made them up. It was Frege, Russell, Quine who had reservations on it even if didn't oppose to it. Wittgenstein sounds he was against it.

You said, "Problem with Set Theory is that their concept "infinite" means "finite""

What set theory textbook, or any reference in set theory or mathematics, says that 'infinite' means 'finite'? — TonesInDeepFreeze

I have already quoted from Wittgenstein from his writings "infinite" in math means "finite", and he adds that the mathematicians discussions will end. It is obvious you have not read the post.

But my point is not about "infinite" is "finite" or whatever. My point was that the concept "infinite" means something totally different, and math's infinity in set theory doesn't exist. This is not what other folks says, or may some folks did, I don't know. But that is just my idea. I don't need any supporting comments on that from anyone, when I think that is the case.

But you are quoting from the old and outdated mathematician Dedekind on the concept of "infinity", and it means "not finite". To me it just sounds vacuous word game to say infinity is not finite, but "not finite". It is a concept which doesn't exist in reality. It is an abstract concept for describing motions, actions and operations.

Anyway, Dedekind's set theory had faults and limitations. Here is what ChatGPT says about his Set Theory and concept of Infinity.

"Dedekind's set theory, while foundational and influential, does have some limitations and criticisms. Here are a few:

Axiomatic Foundation: Dedekind's set theory lacks a formal axiomatic foundation comparable to other set theories like Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Without a clear set of axioms, Dedekind's set theory may be seen as less rigorous or formal by contemporary standards.

Treatment of Infinity: While Dedekind made significant contributions to the understanding of infinity, his treatment of infinity in set theory may be considered less systematic compared to later developments, such as Cantor's work on transfinite numbers and ZFC set theory. Some critics argue that Dedekind's definition of infinite sets as those that can be put into one-to-one correspondence with proper subsets of themselves is not as precise or comprehensive as later formulations.

Lack of Explicit Axioms: Dedekind's set theory does not provide a set of explicit axioms like those found in ZFC set theory. This lack of a formal axiomatization can make it difficult to establish the foundational principles of Dedekind's theory and to reason rigorously about sets within this framework.

Scope and Development: Dedekind's set theory was developed in the late 19th century and may be seen as lacking some of the conceptual developments and formalizations that occurred in later set theories. While his work laid important groundwork for the development of modern set theory, it may not encompass the full range of concepts and techniques found in more contemporary approaches." -ChatGPT

I would have expected your reply to my question from the reputable and well known modern math textbooks which says "infinite" is "not finite", as you have been insisting as the case. But it doesn't matter. To me, infinity is an abstract concept which has no entity, and shouldn't be used for naming the set elements or sets. It doesn't reflect the reality accurately, and is a vacuous concept. Infinity only makes sense when it is describing motions, actions or operations. Or it can be used in the poetry or metaphor as a figure of speech. That is fine.

I am not claiming anything on the math theory. I am just pointing out the contradictions and false information in your posts, and replying to them. It would be a gross distortion of the fact and over exaggeration to state anything more than that about my replies.

yeah, that must be it. — Banno

:nerd: Be honest to yourself, and try to be your own man. :cool:

Apparently, people will also try to do mathematics without the mathematics. — Banno

No one was doing math here. This is philosophy forum, not math. We have been just pointing out that misuse of concepts and definitions, and using them as the premises in their arguments can mislead people with the wrong answers and absurd conclusions.

Pointing out their errors simply makes them double down. Sometimes all you can do is laugh and walk away. — Banno

You claim that you care about philosophy, but don't appear to be doing so. What you seem to be doing here is just codon blindly whoever is on your side whether right or wrong, and laugh and walk away from truths.

Goodness is the property that ascribes whether or not something is moral or immoral, not vice-versa. — Bob Ross

Please prove how Goodness is the property of moral or immoral.

The OP argues that moral goodness is actual perfection, which is self-harmony and self-unity. — Bob Ross

Is there such a thing as moral goodness as actual perfection? Goodness for who? An act is either moral or immoral on the basis of many different factors related to the act and the agents. But where does goodness come from? What is moral goodness as actual perfection?

Don't have time for all the replies I want to make lately, but this one is easy:

Problem with Set Theory is that their concept "infinite" means "finite"
— Corvus

A common definition of 'infinite' in mathematics is 'not finite' You have it completely wrong. Would that you would not persist in posting falsehoods. — TonesInDeepFreeze

Me neither. But I try to reply to the posts addressed to me.
Which math textbook says "infinite" means "not finite"?

It seems you use "perceive" were you might better use "interact".

That might be all that is problematic with this thread. — Banno

Interact? Why do you want to talk, share and communicate with your cup?
We are interested in perception and belief, not interaction.

Goodness is not normative: it is the property of having hypothetical or actual perfection. Normativity arises out of the nature of subjects: cognition and conation supply something new to reality—the assessment of or desire for how things should be (as opposed to how they are). Moral goodness, for example, is just the state of being in self-harmony and self-unity: it does not indicate itself whether something should be in that state. It is up to subjects to choose what should be, and a (morally) good man simply chooses that things should be (morally) good. — Bob Ross

I am just a reader not a philosopher, so most of my views are likely to be from the common sensical ideas. But isn't moral goodness a superfluous term? Why not just say, moral or immoral, instead of moral goodness and moral badness?

Good is too wide term which is usually applied to the situations and things in daily life of the ordinary people. How are you? I am good. How was your weekend? It was good.
Where are you off to this weekend? To a party. Have a good time.
Buy some fish. Try to get good ones.

But morally good? It sounds unclear. Is there such a thing or situation as morally good? Good for who? Isn't just being moral enough?

Similarly, the word "infinity" has one meaning in a formal set theory and a different meaning in everyday natural language — RussellA

Problem with Set Theory is that their concept "infinite" means "finite". It breaks the most fundamental principle of Truth.

In any case, I remember this thread being about solipsism, but it seems the OP was edited to mean something more like object permanence and the problem of induction, or perhaps I misread the first time. — Lionino

The OP was not edited at all. But due to misunderstanding of many folks in their posts (including Banno), there had been extra posts added by me for clarifying and broadening the OP into any possible exploratory discussions on the elementary concepts in the OP title as well as general epistemological, sceptic, ontological and logical issues in perception.

But it's not a belief. The world really exists. And it really exists precisely because there is nothing outside of ideas or perceptions. Since there is nothing outside those, there is no "outside" at all, and since there is no outside, the so-called "inside" is actually the world itself. So the world does exist. It lies within the idea itself. Idealism leads to realism and realism leads to idealism. It's a "loop". — LFranc

When one is a hard idealist, and the world is just a representation in his mind, it would be hard to refute him. Indeed if what you see is a representation of the world, how do you know the real world?
If you are a part of the world, do you even exist yourself?

Well yes, there are good reasons to doubt that the cup will remain in the cupboard. The point here is simply that your "when I am not perceiving the world, there is no reason that I can believe in the existence of the world" is not a good reason to think that the cup has disappeared from the cupboard. — Banno

When you are not perceiving the world, you wouldn't be asking the question where is my cup, would you? The question sounds absurd.

This had me puzzling. How do you go about buying coffee? There's the package on the shelf at the store, brightly labeled "Dark Roast". But when one is not perceiving the coffee, — Banno

Again when you are not perceiving the world, you wouldn't be going out buying coffee either. Isn't it an absurd puzzling? The puzzle must be an illusion when you are not perceiving the world. Where does your puzzle come from?

No matter what the textbooks say, one must be able to ask Why? instead of just blindly accepting the answers and claim that it is the only truths because the textbooks say so.
— Corvus

Again, you are unfamiliar with any of this; you are blindly punching. — TonesInDeepFreeze

That was an accurate description of the problems of the mathers. Not blindly punching anything at all.

The other shortcomings of math is it cannot accurately reflect the real world and its problems. It often distorts it via the unfounded and unjustified concepts and axioms, hence arriving at nonsense.

Mind you when math started in ancient Egypt, it used to be for mainly the practical problem solving purposes e.g. counting the sheeps, cows, and apples in the markets, and finding out the boundaries and locations for the pyramid locations in the deserts.

It used to work well, but once math started running the blind free rein of modifying the abstract concepts and keep deducing the illusional theories, things started going wrong turning the empirical and pragmatic skills in origin into some sort of an abstract subject which sometimes speaks in the tone of deeply frozen religion. Not cool at all.

As I said, there are deep, puzzling questions about mathematics, but that doesn't make the mathematics itself, especially as formalized, confusing. On the contrary, if you ever read a treatment of the axiomatic development of mathematics, you may see that it is precise, unambiguous, objective (in the specific sense I mentioned), and with good authors, crisply presented. — TonesInDeepFreeze

Axiomatic methodology in math is not free from problems and deficiencies. They are subjective definitions which are often circular in logic. They lack in consistency and are incomplete in most times.
They are dependent on the other axioms mostly. Most of them are abstract and illusional which renders to the false conclusions. A typical example is the Infinity in Set theory.

Perhaps not axioms as the main approach. And philosophy ranges from poetic through speculative, hypothetical, concrete and formal. But deductive reasoning and demonstration is basic and ubiquitous in large parts of philosophy. And the axiomatic method does appear in certain famous philosophy, and its principles and uses - sometimes even formalized - are prevalent in modern philosophy, philosophy of mathematics and philosophy of language. — TonesInDeepFreeze

Sure as endeavours to be formal and more clear in their system, but is it always making sense? That is another question. Often it tends to make the system look more convoluted, if not done properly.

But when I mentioned objectivity, of course I was not referring to objectivity of philosophy, but rather the objectivity of formal axiomatics, in the very specific sense I mentioned. And that is a philosophical consideration. Then you challenged my claim that mathematics has that objectivity. So I explained to you again the very specific sense I first mentioned. The fact that philosophy in its wide scope is not usually characterized as axiomatic doesn't vitiate my point. — TonesInDeepFreeze

Objectivity is the objectivity of knowledge. Not objectivity of philosophy or objectivity of mathematics. That is another misunderstanding of yours. I wouldn't be surprised if you go on claiming an objectivity for set theories and an objectivity for numbers ... It is like saying a subjectivity of objectivity. A contradiction.

What passages from Frege, Russell or Hilbert do you have in mind? — TonesInDeepFreeze

You must read them yourself. They all had reservations on the concept of Infinity in math. Quite understandably and rightly so.

↪Corvus
I do believe in the existence of the cup when I am perceiving it, but when I am not perceiving it, I no longer have a ground, warrant or reason to believe in the existence of it.
Indeed, and this is what Berkeley said. Something that would exist independently of a perceiving mind is unverifiable. Because, if you check that such a thing exists, well, too late, you're using thought again. That is the powerful argument by Berkeley. — LFranc

The point at the time of writing the post was logical ground rather than physical, ontological or epistemic ground for the doubt. If your ground for believing in the world is your perception (P), then
what is the ground for the belief when not perceiving the world? (¬P).

It wasn't about the existence of a cup, or any particular physical objects as such. It was rather about the the nature of our belief in the existence of the unperceived objects or world.

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 might had been a situation where you put the cup in the cupboard of the shared kitchen dormitory in your university time. I wonder if you had ever lived in a dormitory of a university with the other folks sharing a kitchen. I had long time ago.

A cup can go missing in shared kitchen cupboard like that with other folks coming into the kitchen and grabbing whatever cup they see when they open the cupboard, make coffee and take it to their room. This used to happen often, and I had to look for an any free cup for making coffee for me.

If you were buying some coffee for yourself in a spar, and see new cups for a dollar or two beside the coffee jars in the shelf, then you might decide to buy them because you doubt if your own cup in the cupboard has been taken away by some other folks in the corridor, and you will never see it again. Yes, you might doubt if your cup exists or not. Why not?

In real life, people move things around, buildings and houses get demolished for new development, roads and grounds get eroded by heavy rains, trees get chopped off, people born, people die, people leave, the sun keeps rising and setting, and time passes non-stop. Nothing remains the same. Why should you stop doubting? If you don't doubt, that's not right.

In information science, an ontology encompasses a representation, formal naming, and definitions of the categories, properties, and relations between the concepts, data, or entities that pertain to one, many, or all domains of discourse. — PL Olcott

How does your system deal with the same words of the different meanings in the real world identification?
For example, a dog is an animal. But you also get a dog which has the following meanings.
1. A dog as a workbench tool.
2. A dog as a worthless or contemptible person.
3. Any of various usually simple mechanical devices.
4. The astronomical constellations - Canis Major or Canis Minor.
5. An an inferior one of its kind.

the whole picture was based on the fabricated concepts, which are not very useful or practical in the real world.
— Corvus

Fabricated in the sense of being abstract. And it is patently false that classical infinitistic mathematics is not useful or practical. Reliance on even just ordinary calculus is vast in the science and technology we all depend on. — TonesInDeepFreeze

"A careful reader will find that literature of mathematics is glutted with inanities and absurdities which have had their source in the infinite. " - David Hilbert, On the Infinite, pp.184 Philosophy of Mathematics Selected Readings, Edited by H. Putnam and P. Benacerraf 1982

Not just because it's what a book says. Rather, textbooks provide proofs of theorems from axioms (including definitional axioms) with inference rules. One doesn't have to accept those axioms and inference rules, but if one is criticizing set theory then it is irresponsible to not recognize that the axioms and inference rules do provide formal proofs of the theorems. Moreover, intellectual responsibility requires not misrepresenting the mathematics as if the mathematics says that the theorems claim simpliciter such things as that there are infinite sets of physical objects or even that there are infinite sets in certain other metaphysical senses of 'infinite' — TonesInDeepFreeze

The textbook axioms and formal proofs of the theorems are subject to change or found out to be falsity at any moment when someone comes up with the newly found axioms and proofs against them. In that case it would be the one who used to think that their claims were the truths, have been actually spreading misrepresentation of the knowledge. No matter what the textbooks say, one must be able to ask Why? instead of just blindly accepting the answers and claim that it is the only truths because the textbooks say so.

Bottom line is that, truth speaks for itself. One doesn't need to say to the others, they are wrong unless when it is absolutely necessary. But just tell the arguments and conclusions, which are true. If in any case of doubt, ask why and how so.

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.