Look at the links and my remarks about them.

Yes, he made his post sounding like that. Do you still not understand any simile or metaphor expressions in English? — Corvus

That is ridiculous. You accused him of exaggeration. That's not simile or metaphor. So you exaggerated, not him.

See your ad hominem attacks on other interlocutors from the beginning of your posts? That is not a good manner at all. Please just discuss the philosophy. Have some respect. Don't throw insults to the other interlocutors. — Corvus

First, you lied that earlier I began with ad hominems.

Second, you skipped my reply about that my arguments are not ad hominem, but rather I give arguments that are not ad hominem but also add the needed observation that the interlocuter is indeed ignorant, confused, and dishonest, as at a certain point it deserves remarking that he is ignorant, confused and dishonest.

Indeed, you can see that my first posts in this thread, and others like it, are devoid of personal remarks, and my first posts to new interlocuters are devoid of personal remarks. But, eventually stubbornly ignorant, confused and dishonest posting deserves to be pointed out for being what it is.

Meanwhile, as you take such umbrage to disparaging remarks, you're free to cut them out of your own posting. I think it's your prerogative to make them, but yours happen to be quite inapposite, which is putting it mildly.

Still speaking on behalf of Banno? — Corvus

No, I don't speak on his behalf. I speak on my own behalf to say that it is a plain fact that Banno did not exaggerate by saying 'hundreds' but that you exaggerated by saying that he did say 'hundreds'.

Again it's in the plain record of the posts.

Where is your logic and evidence for your claim? — Corvus

I posted the links. That's the evidence. The logic is pretty much inferring that what is posted at the links says just what it says.

Order has nothing to do with this.

An ordering is a certain kind of relation on a set.

The axiom of extensionality pertains no matter what orderings are on a set.

{0 1 } = {1 0}

{<0 1>} is an ordering on {0 1}
and
{<1 0>} is an ordering on {0 1}

{<0 1>} not= {<1 0>}

{<0 0> <1 1>} is a sequence whose range is {0 1}
and
{<0 1> <1 0>} is a sequence whose range is {0 1}

{<0 0> <1 1>} not= {<0 1> <1 0>}

Orders and sequences are rigorous in set theory. And the axiom of extensionality is not inconsistent with them.

Any time. Thank you.

The crank is ignorant and confused about identity theory and the axiom of extensionality, so this at least is reference for how they actually work in set theory:


An axiomatization of identity theory:

Axiom:

Ax x=x

Axiom Schema:

If P is an atomic formula
and
Q is the same as P except y occurs in zero or more places in Q where x occurs in P

then all closures of the following are axioms:

(x=y & P) -> Q

/

Set theory has the primitive 2-place relation symbol 'e'.

We can formulate set theory with the 2-place relation symbol '=' taken as prmitive, or we can formulate set theory with the 2-place relation symbol as defined. The formulations are equivalant.


Primitive:

Adopt the axioms of identity theory and add the axiom of extensionality. But in the language of set theory, the only atomic formulas are of the form xez for any variables x and z (so also zex, yez and zey).

So the axioms of identity theory for the language of set theory include:

Ax x=x

Axyz((x=y & zex) -> zey))

Axyz((x=y & zey) -> zex))

Axyz((x=y & xez) -> yez))

Axyz((x=y & yez) -> xez))

Axiom of extensionality:

Axy(Az(zex <-> zey) -> x=y)

So we have the theorem:

Axy(x=y <-> (Az((zex <-> zey) & (xez <-> yez))


Defined:

Axy(x=y <-> (Az((zex <-> zey) & (xez <-> yez))


So the two formulations are equivalent.


Notice that we have these theorems:


Axy(x=y -> (Az((zex <-> zey) & (xez <-> yez)), which carries, for the language of set theory, Leibniz's indiscernability of identicals.

and

Axy((Az((zex <-> zey) & (xez <-> yez) -> x=y), which carries, for the language of set hteory, Leibniz's identity of indiscernabiles.

/

Moreover, the consistency of the axiom of extensionality with idenity theory is trivally proven by the trivial model I mentioned:

The domain of discourse is {0 {0}}

'=' for the identity relation on the domain of discourse, i.e., {< 0 0> <{0} {0}>}

'e' for the membership relation on the domain of discourse, i.e., {<0 {0}>}

In that that model, all the axioms of identity theory and the axiom of extensionality are true.

The crank can't properly respond to that, because he doesn't know anything about models and consistency.

I really hope you get well soon.

The leap is in the form of axioms.

Just to be clear, the logic system itself is not contravened. Rather, we add non-logical axioms. We add axioms that are consistent, thus true in some models, but that are not logically true, thus not true in all models.

Don't know that book, but

Ax x*0 = 0 is an axiom of first order PA, so it's easy to prove x*0 = 0

and in set theory, the PA axioms are theorems.

/

It was wonderful me in the 5th grade to learn about reasoning in mathematics and seeing things like different base numbering systems and modular arithmetic rather than just memorizing multiplication tables, executing steps in long division and reducing fractions to lowest terms.

I read most of 'Fashionable Nonsense' quite a while ago. There were quoted examples from certain writers. If those quotes were in fair context, then indeed those writers are completely full of BS regarding the mathematics they mentioned.

Mathematical logic at least explicates symbolic logic, and symbolic logic is useful. We are all typing on computers whose invention and development are based on concepts in symbolic logic, mathematical logic and the theory of computability that really took off with mathematical logic (though, I woudn't necessarily be unsympathetic to the idea that we might all be a lot better off without these blasted, annoying, buggy, and intentionally mal-designed digit boxes).

Another episode from 'The Adventures of the Crank Metafizzled Blunderbound in the Land of Daily Life':

Support Rep: Hello, thank you for calling MetaCard, your card to be used anywhere for making purchases of things about things. How can I help you today?

Crank: It says on my statement that I have 10000 rewards points. Please send them to my address: 666 South Georg Wilhelm Friedrich Hegel Alley, Apartment 0, Unreality Village, Planet Mars. No zip code on that.

Support Rep: I'm sorry, but they are rewards points. We don't send them to you. You use them for discounts on things.

Crank: How can I use them if I don't have them here? If they're real, then you can send them to me.

Support Rep: They are real. But they're not like physical goods that we ship to you.

Crank: Please don't tell me that you too have been infected by set theory! I'm asking you again to send me my points!

Support Rep: I'd like to help. Let me explain it this way: The rewards points are a number. It is noted on your account. Your points may change, but when we say "Metafizzled Blunderbound's points on February 18, 2024" we mean whatever number of points is on your account at that date. So it's a number, not a physical thing. However, depending on what that number is, you may apply that number for discounts on purchases you make. For each point you get ten cents off your puchase. There's nowhere for me to go get your number of points and ship them to you.

Crank: Look, what is that number now?

Support Rep: Metafizzled Blunderbound's number of points is 10000.

Crank. No, no, no! Not is. Equals! The number of my points equals 10000. Not the number of points is 10000. It is nonsense to say that the number of my points is 10000!

Support Rep: I can say 'equals' if you like. But I'm just telling you what your number of points is.

Crank: You are talking nonsense, and these are not even real things, so don't even bother!

Support Rep: So you don't want to apply your points to any purchases?

Crank: How can I apply them if they're not even real?! I surely am not going to be fooled into trying to use something that is not real! And I definitely am not going along with your bad philsophy where you say is instead of equals! Good day to you! (Hangs up.)

Two weeks later:

Bob (Blunderbound's neighbor): Hey, Blunderbound, how's it hangin'?

Crank: Okay, Bob. You should see how I'm tearing apart those set theory guys at The Philosophy Forum. Ha, I even showed them that Chat GPT says I'm right!

Bob: That's great. Hey, check out this waterproof watch I got. Got a discount with my MetaCard points. Have you used any of your points?

Crank: No, I don't use things that are not real. Besides, I don't need a watch. I have my sundial in the backyard. (Music cue, trombone "wamp wamp wamp")

Still picking up after the crank's daily littering:

The axiom of extensionality is not used to prove axioms. Rather, the axiom of extensionality is used to prove theorems.

/

The denotion of '1+1' is not truth by correspondence. '1+1' is not a statement, thus it doesn't have a truth value.

/

There are non-platonistic senses in which we may speak of mathematical objects.

/

Identity theory is congruent with Leibnitz's principles of the identity of indiscernibles and the indiscernabilily of identicals.

/

Chat GPT is not even remotely a reliable reference. Even a lowly fortune cookie can tell you: "He who cites Chat GPT as authority makes of himself a fool". Anyway, just to see, I asked Chat GPT, and it said that 1+1 is 2, and it did not confine that to set theory. And, of course, the cranks SKIPS that. Anyway, that a supposed philosopher is citing the famously "hallucinating" Chat GPT as an authority indexes how pathetic the Internet has become, not merely how pathetic he has always been.

Which is to say, he cannot do the impossible.

The cranks says, "Tones told you 'don't to listen to that machine it doesn't support me', or something like that."

Again, the crank LIES about me.

I never said that Chat GPT is not to be trusted because it doesn't agree with me.

Rather, it is not to be trusted because, over and over and over, it has been demonstrated, by many people, to spout blatant falsehoods. Indeed, even the makers of AI themeleves stress that AI is not necessarily a source of information but rather its primary role is as a composition tool.

Anyone can see for themselves that Chat AI outright fabricates, easily by asking it questions that it does not have ready answers to.

Moreover, in my conversation with Chat AI, it did say "1+1 is 2". But the crank SKIPS that.

To advance his illogical and confused argument, the crank resorts to LYING about what I said. He is pathetic.

Infinity is unknowable by the finite human mind, yet we know the meaning of "infinity" — RussellA

In mathematics, there are 'points of infinity' (which are not necessarily infinite sets), and the expression 'to infinity' (which is a figure of speech that when we unpack it we don't have an 'infinity' that is referred to), etc. But ordinarily there is no object that we name with the noun 'infinity', rather there is the adjective 'is infinite'. So, at least in a mathematical context, "Infinity is unknowable" doesn't have an apparent meaning to me. On the other hand the meaning of 'is infinite' is quite clear, as it means 'not finite'.

Abstract concepts don't of necessity refer to physical things, but wouldn't exist without physical things — RussellA

That requires a framework that defines 'physical thing' or takes it as primitive. Then, depending on whatever framework is proposed, we could examine whether there cannot be abstractions independent of physical things.

It does seem to me that concepts are formed from prior ostensive inferences. But that is epistemological, not necessarily ontological.

I'm just glad that when I think of mathematics, there is the abstract mathematical object 1 and that there are not as many of the number 1 as there are each of certain physical events in brains starting and stopping.

Along the lines of "Bad publicity is at least publicity".

The axiom of extensionality is not inconsistent with identity theory.

Here is the most trivial model of both the axiom of extensionality and identity theory ('e' for the epsilon symbol):

The domain of discourse is {0 {0}}

'=' for the identity relation on the domain of discourse, i.e., {< 0 0> <{0} {0}>}

'e' for the membership relation on the domain of discourse, i.e., {<0 {0}>}

In that that model, all the axioms of identity theory and the axiom of extensionality are true.

In this part of the discussion, as in some recent posts, that is not directly about what mathematics itself in fact says, I am not trying to convince anyone else to see it the way I do. Rather, these are explanations of my best attempts to, for myself, have a framework to understand abstraction, truth and related questions. As a very broad generalization, I think of at least these two categories: (1) Matters of fact. (2) Matters of frameworks for facts. With (1), truth and falsity apply. With (2), coherence and explanatory robustness applies.

In this thread, for example: That ordinary mathematics says "1+1 is 2" is matter of fact. But whether ordinary mathematics should say that 1+1 is 2 is a matter of framework. And I take ordinary mathematics not as an account of facts about concretes, but rather as a framework for such facts. And, again, we need to distinguish between what ordinary mathematics does in fact say with what one may think it should say and what one thinks should be the framework for mathematics.

Frameworks involving abstractions sometimes work in an "as if" way. There is a wide range of thinking about what mathematical objects are - platonic, fictional, in some special sense concrete, or even extreme nominalism. But whatever we take mathematics to be talking about, at least we may speak of abstractions "as if" they are things or objects. Not concrete objects, but "as if" they are handled grammatically similar to they way concretes are handled.

Common examples, to the point of cliche, abound, such as that the knight in chess is not a concrete knight, not any particular piece of wood or plastic resting on a particular piece of wood or cardboard, but rather an abstract concept that we speak of similarly to the way we speak of concretes. This similarity does not imply that the abstract chess object that is the knight is a certain piece of wood. And so we use such locutions as "It either moves up or down one square vertically and over two squares horizontally, or up or down two squares vertically and over one square horizontally" The 'it' there must refer to something, and it sure as shootin' don't gotta a physical object. Also, 'moves' and 'square'. Moves can even be made by telephone and the piece that one player moves on his board at home is a different piece from the one the other player moves on his board at home, but it is understood (courtesy of ABSTRACTION) that those pieces represent the SAME knight. One can even perform a game of chess purely mentally, as chess masters actually do. And just as one can perform arithmetic and even more complicated mathematics purely mentally. That is courtesy of ABSTRACTION.

And the number 1 in mathematics is an abstract mathematical object that we speak of in a similar way to the way we speak of concretes, but that does not imply that the number 1 is a concrete object.

Going back to even more fundamental considerations, my starting premise is that experience is occurring. And as 'experience' and 'occurring' are the notions I start with, I must take them as primitive.

Notice that I didn't say 'experiences' plural, because I had not yet gotten to saying that not only does experience occur but that experience has parts and thus there are a multiplicity of experiences. But I do go on to say that if I do not allow there are many experiences, it would be intractable for me to talk about the experience that occurs.

Notice that I didn't say "I am having experiences", since I had not yet gotten to a premise that there is a thing that is named 'I'. But then I do refer to 'I' and as a thing, as it would be intractable for me to go beyond 'experiences are occurring' without being able to couch my experiences with reference to 'I', thus I as a thing.

As I go on, I find that certain other notions such as 'is', 'exists', 'thing' or 'object', 'same' 'multiple'. etc. are such that I don't see a way to define them strictly from the primitives I've allowed myself. So I then take such notions as themselves "built in" to whatever thinking I'm going to be doing.

So, by adding more concepts, I eventually - pretty crudely, without all the steps filled in, since I don't claim to have provided even for myself a rigorous philosophical system - get to the idea that there are other people having experiences, and that enough of these experiences have commonality among people such that we can submit claims about them to a process of judging claims as publicly factual or not, and with finer and finer standards of judgement such as those of the sciences. But the very determinations of fact, let alone the conceptual organization of facts, are vis-a-vis frameworks, and it is not disallowed that one may use different frameworks for different purposes.

For me, the value and wisdom of philosophy is not in the determination of facts, but rather in providing rich, thoughtful, and creative conceptual frameworks for making sense of the relations among facts. And, again, different frameworks may be used for different purposes.

Meanwhile, I would not contest that formation of concepts relies on first approaching an understanding of words ostensively.

As I say, I do not propose this as prescriptive, but rather only that it describes my own humble attempt to make sense of stuff for myself. Hopefully it might be a heuristically useful for others, but I don't insist that it must be.

I see no philosophy nor mathematics in your latest replies to me. — Metaphysician Undercover

Make an appointment with an ophthalmologist.

Why do you hope that many people want your posts to be monitored by the moderators?

Crank: I want six plus two cans of that delicious abstraction-free metaphysical underground crank juice.

Shopkeeper: Here you go, eight cans of crank juice.

Crank: No, I said I want six plus two cans.

Shopkeeper. But six plus two is eight.

Crank: No, you must be reading those lying fool mathematicians with their extensional identity nonsense. I bet you even recite that awful axiom of extensionality every night before you go to bed. Six plus two means that I want you to get six cans then two more.

Shopkeeper: But I didn't have to do that. I already knew that I had eleven cans on the shelf, so I gave you all the cans on the shelf except three, so I subtracted three from eleven to give you eight cans.

Crank: No! No! No! I told, you just the way I posted at The Philosophy Forum, that six plus two is not eleven minus three. You're taking addition and subtraction extensionally! They are intensional, you dumb cluck! When I tell you what I want, I want it intensionally not extensionally! How can I be any clearer that six plus two is not eight?!

Shopkeeper: I'm very sorry, sir, but do you want the eight cans of crank juice or not?

Crank: Just forget about it. I want six plus two cans, not eight cans and definitely not eleven minus three cans! I guess I'll have to take my business elsewhere, even if I have to drive a potentially infinite number of miles to get there!

That's enough for now in reply to the star crank. It's just too laborious to correct every one of his confusions.

Several years ago, I moved to an apartment across town. Everything was very nice on the grounds of the apartment and in the neighborhood. Except the litter on the sidewalks. The first day, I picked up all the litter in front of my building, thinking that I wouldn't have to pick up for maybe another week or two. But the very next day, there was even more litter than I picked up the day before. And over time I started noticing that it was the same candy wrappers and soda cans every day. So it seemed that much of it was coming from a certain person. Someone daily spewing the same trash.

'1+1' does not equal '2'

'1+1' is not '2'

the denotation of '1+1' equals the denotation of '2'

the denotation of '1+1' is the denotation of '2'

1+1 equals 2

1+1 is 2

One may say that they don't themselves construe that way or that we shouldn't construe that way. But it is counterfactual to say that we don't construe that way.

/

One may claim that abstract objects should not be referred to in identity statements. But thought and communication, not just about mathematics, but even about everyday ideas, pretty much crashes if we are not permitted to apply the identity relation to abstract objects.

/

What passages in Aristotle are being referred to?

/

Saying that '1+1' names an operation rather than the result of the operation with the arguments is merely assertion. Again, one may say that one thinks that '1+1' should be regarded as a name of an operation, but that does not entail that in fact that is now how '1+1' is construed in ordinary mathematics.

/

The order of evaluation of terms and formulas is recursive. There is no ambiguity in that regard.

/

Yes, one can utter "1+1 = 8-1", but if the denotation of '+' is the addition operation on integers, the denotation of '1' is the integer one, the denotation of '8' is the integer eight, the denotation of '=' is the identity relation, and the denotation of '-' is the subtraction operation on integers, then "1+1 = 8-1" is false. This in no way vitiates that '=' stands for the identity relation. On the contrary, it is an example of the mathematics working just as we want it to work, just as it works even for the crank when he adds the number of pens on his desk, figures out his finances, relies on the entire body of science that allows him to stay alive, and uses the very computer he types on to send his ignorant, illogical and confused posts to this forum.

\

If one wishes to take '+' as intensional and not extenstional, then one should have a ball doing that. But that doesn't require that ordinary mathematics does that, espcially as, without some alternative rigorous framework, it would render thought and communication about even everyday mathematics unmanageable.

Your problem is that you (Banno) blindly say that others' points are wrong before presenting your arguments with evidence supporting your claims — Corvus

He did not say that anyone is wrong merely by the fact of disagreement.

My point was simple, and I quoted one philosopher, from which was the Wittgenstein's — Corvus

And that quote doesn't support the claim you made that mathematics regards 'infinite' as meaning 'finite'. Though, lately, you say that claim is only a metaphor for something. If you like, you may remind me what exactly you intend it to be a metaphor for.

I have not made many claims quoting hundreds of philosophers. That is just another distortion of the truth with exaggeration. — Corvus

The quote above, written to Banno, is exaggeration thus distortion.

Banno didn't say that you have made claims by quoting hundreds of philosophers..

The agent can start to predict what would happen if marks were modified this way or that. I would say that once an agent starts this sort of imagining, it has started thinking mathematically. — GrahamJ

That's an interesting idea.

Many would believe that your (Banno's) posts should be under proactive moderation — Corvus

Who are those many people?

Anyone thinking differently from you (Banno) are downright wrong, and misattributing. — Corvus

Did Banno ever say or imply that he believes that?

I wasn't meaning to say the book is denying, accepting or defining on the infinity as per my view. — Corvus

You're confused as usual. I didn't say anything about your view of infinity regarding the book. Rather I note that you challenged me to show you a book in which 'infinite' is defined as I said it is defined in mathematics, while that definition is in the very book you mention as your reference. Again: It's not a matter of whether you agree or not with the definition, but rather that the definition is given in that book while you challenged me to cite such a book.

Meanwhile, my point stands that I think the chapter on the history and philosophy discusses the very point I made a while ago, but which you rejected, about the benefits of axiomatization.

one that you must read about is "Quine". — Corvus

I have read Quine. Not enough though, since logic and mathematics not topics to which I devote my primary attention. If there is something you have to say about Quine, then you can say it.

If I really lied, then I would have told you that I lied, which is true. — Corvus

That is false, since you didn't say that you lied but you did lie.

The plain record of the posts in this thread prove that you lied, as I explicitly linked to the posts. But you skip that.

Putnam edited a book called Philosophy of Mathematics Selected readings. He put in there various articles by different people. — Corvus

When you said that you base on philosophy of mathematics and mentioned Putnam in particular, naturally I thought you meant that you base on views of Putnam. From what you said, one couldn't be expected to think that what you actually meant is that you have read a particular book edited by Putnam.

But now that you have added to your original statement, fair enough, your views are informed by reading that book.

So now, what are the articles in that book that you base your views on?

That book is full of great stuff. One article in particular that I think helps a lot is Boolos's 'The Iterative concept of set'.

You obviously have no idea about the book, or what the Edited book means. — Corvus

It is a really stupid inference from (1) I took you to mean that you base your views on Putnam when you said you base on philosophy of mathematics and mentioned Putnam in particular to (2) I don't know what an edited book is.

My point is that I agree that it is not the case that an abstract concept corresponds to one particular concrete instantiation, but rather we can only understand an abstract concept by thinking of some concrete instantiation of it — RussellA

That deserves consideration, though I'm not sure about it while also I don't have an argument in disagreement to give at this time.

What is a concrete example of the concept of 'does not exist'? What is a concrete example of 'there are things that do not exist irrespective of any list of properties that no existing thing have'?

Yet, the notion of 'concrete instantiation' is itself an abstraction made of the the two abstractions 'concrete' and 'instantiation'.

Also, the lines I'm thinking along is that certain utterly basic abstractions, such as 'object', 'thing', 'entity', 'is', and 'exists' themselves presuppose abstraction no matter what concretes are involved or not.

Anyway, to say that thinking of abstractions requires thinking of a concrete examples does not say that we don't think of abstract objects; or at least that to demand that we utterly eschew a notion of abstract objects brings even everyday conceptualization to a screeching halt or at least a devastating slowdown.

Doesn’t it require interpretation? — Joshs

One may discuss its philosophical implications, but the proof itself doesn't require a philosophical interpretation.

I am not familiar with the notion of 'relativizing its very nature', so I can't opine on it.

It's just philosophy here not pure math. — Mark Nyquist

I have addressed that so many times in so many threads. Maybe earlier in this thread too.

Got it. Thanks.

I say that without prejudice to the question of whether the mentioned postmodernist philosophers do or do not understand mathematics as well as Descartes and Leibnitz did (even recognizing that Leibnitz's calculus needed to be rectified by late 19th century concepts and then 20th century axiomatizations (which also include non-standard analysis that does formalize infinitesimals)).

But do you think those postmodernist philosophers understand 20th century foundational mathematics as well as mathematicians and certain others in the philosophy of mathematics do?

How one regards the significance of formal proof and formal theories may be philosophical, but the incompleteness proof itself about formal theories does not require any particular philosophy.