If x, y, and z are sets that are members of themselves, and I form a set of these three sets, to represent this, I can write something like: p = {x, y, z}. I cannot write x = {x, y, z}. — Philosopher19

That is incorrect.

Of course, to countenance sets being member of themselves, we have to delete the axiom of regularity. With that done:

Suppose xex, yey, and zez

Suppose x = {x y z}

Those suppositions together are consistent. For example by letting:

y = {y} and z = {z}

You cannot have a set of ALL sets that are not members of themselves — Philosopher19

With the axiom of regularity, that is correct, since such a set would have all sets as members, from which we derive a contradiction.

The Russell principle though is:

There is no set of all and only the sets that are not members of themselves.

And that is derived from first order logic alone:

For any relation R, ~ExAy(Ryx <-> ~Ryy)

You cannot have a set of ALL sets that are members of themselves — Philosopher19

That is incorrect. With the axiom of regularity, that set is the empty set. And without the axiom of regularity, it would still be consistent for there to be a non-empty set of all sets that are members of themselves. For example, allow that there is just one set S that is a member of itself. Then the set of all sets that are members of themselves is {S}.

the version in the video — TheMadFool

If you can correctly extract from the video that Godel's argument is circular, then the video is wrong. We should learn Godel's argument from a carefully written exposition, not from a merely breezy cartoon version.

Get your hands on an introductory course on logic. — TheMadFool

What is the introductory course you have taken?

I see you use some sentential logic, but a good understanding of Godel's theorem requires also predicate logic, some set theory, and a first course in mathematical logic.

(By the way, in a conversational context, spelling out and numbering, as you do, such basic sentential logic as applications of modus ponens and conjunction is gratuitous pedanticism that only clutters up whatever it is you mean to say. People don't need to have such basic reasoning annotated for them.)

Godel uses the liar paradox to wit, the sentence L = This sentence is false — TheMadFool

That is flat out incorrect. Godel uses an argument analogous to the liar paradox, but not the liar paradox and nothing like "this sentence is false". Rather, it mathematically renders "this sentence is not provable in system P".

You are spreading disinformation, Porky.

Is this what you were referring too? — Gregory

I told you twice what Godel's theorem is:

If a theory T is a consistent, recursively axiomatizable extension of Robinson arithmetic, then there is a sentence G in the language for T such that neither G nor ~G is a theorem of T. — TonesInDeepFreeze

Instead of recognizing that, you bring up a different matter.

You will not make any progress here if you can't pay attention.

You have so many misconceptions. But let's take one thing at a time, starting here: Do you recognize that I did state Godel's theorem?

correction: When I said 'total linear ordering' I should have said 'strict linear ordering'.

You don't seem to know how to read very well. — Metaphysician Undercover

I read fine. But with your lack of replies to many crucial points, I admit that I can't read what doesn't exist.

Here is a previous post tracking your recent evasions:

https://thephilosophyforum.com/discussion/comment/545568

And now the points in this post too:

https://thephilosophyforum.com/discussion/comment/545558

Morris Kline's book, Mathematics: The Loss of Certainty. — fishfry

That book gets important technical points wrong and it's a deplorably tendentious hatchet job. (I don't have the book, and it's been a long time since I read it, so I admit I can't supply specifics right now for my criticism.)

even for you Gödel's theorem is hard to put into words — Gregory

What? It's not at all hard for me. I did it a few posts ago!

If a theory T is a consistent, recursively axiomatizable extension of Robinson arithmetic, then there is a sentence G in the language for T such that neither G nor ~G is a theorem of T. — TonesInDeepFreeze

Please let me know that you see it now so that I may know that I'm not posting to an insane person. Then I'll see about correcting yet more of your ignorant confusion in your post above.

To repeat, since you skipped this:

I don't believe in self reference in math or logic
— Gregory

You don't know the actual nature of the "self-reference" in Godel's proof. The proof may be formulated in finite combinatorial arithmetic. If you have a problem with the proof, then you have a problem with finite combinatorial arithmetic. — TonesInDeepFreeze

But [Russell's] paradox is false. — Gregory

Every contradiction is false in every model. So what?

Meanwhile, it is a theorem of first order logic that there is not an x such that for all y, y bears relation R to x if and only if y does not bear relation R to y.

A set containing itself is just bizarre — Gregory

So what? Neither Russell nor Godel depended on a claim that there is a set that is a member of itself.

What I said about Gödel was based on what the majority of people have said — Gregory

As Seinfeld put it, "Who are these people?" Whatever "majority of people" you talked with, your conversations did not supply you with a even a fraction of a decent understanding of Godel's theorem.

Someone needs a really good background in math to read his actual papers — Gregory

His original papers are rather old-fashioned in their notation. More recent textbooks have pedagogically supplanted the original papers.

so most of us are getting our ideas from second hand sources — Gregory

Since I don't know your sources, I can't say whether the fault is in the sources or in your misunderstanding of them.

you can't prove that a set can contain itself from math itself — Gregory

In set theory, we prove that there is not a set that is a member of itself.

However, with set theory without the axiom of regularity, there is not a proof that there is not a set that is a member of itself.

And, dropping regularity, but adding a different axiom, there is a proof that there is a set that is a member of itself.

so rejecting Russell's paradox is a good way to start in approaching Godel. A set containing itself IS self reference — Gregory

Those two sentences alone are proof that you are completely mixed up and ignorant of what Godel's theorem is.

/

To understand this subject properly, one should learn basic symbolic logic, then a small amount of basic set theory, then an introductory course in mathematical logic - either in a class or by self-study.

Meanwhile, the best book about Godel's theorem for everyday readers:

Godel's Theorem: An Incomplete Guide To Its Use and Abuse - Torkel Franzen.

That book will disabuse you of your confusions.

Imagine some unprovable proposition. Can it be *understood* intuitively like axioms are and be taken as axioms? — Gregory

That is a question that could be asked only by someone unfamiliar with the basics of this subject.

If P is a closed formula, then there is a system S such that P is an axiom for S.

I don't believe in self reference in math or logic — Gregory

You don't know the actual nature of the "self-reference" in Godel's proof. The proof may be formulated in finite combinatorial arithmetic. If you have a problem with the proof, then you have a problem with finite combinatorial arithmetic.

You did all that and managed still not to know what Godel's theorem is.

why haven't you written a couple paragraphs here saying what Gödel really did — Gregory

It's not required for pointing out that you don't know what Godel's theorem is.

But I will indulge you:

The Godel-Rosser theorem may be given a modern statement as:

If a theory T is a consistent, recursively axiomatizable extension of Robinson arithmetic, then there is a sentence G in the language for T such that neither G nor ~G is a theorem of T.

For a video such as this, the very first words should be:

"I'm going to give you an extremely simplified version of some very complicated mathematics. These simplifications gloss over crucial technical details; thus the simplifications may be misleading if one does not at some point go on to understand the actual mathematics. So, we must be extremely careful not to extrapolate philosophical conclusions from our very cursory treatment of this technical subject."

It basically just boils down to how any language capable of formulating e.g. a proof of arithmetic is also capable of formulating self-referential sentences to which there cannot be assigned only one or the other boolean truth value: they must be assigned by the language either neither truth value (so the language is incomplete) or else both truth values (so the language is inconsistent). — Pfhorrest

That is not at all a reasonable summary of Godel's theorem. Just to start: languages are not what are complete or incomplete, but rather theories are complete or incomplete. Also, it is crucial to understand that Godel's theorem has a purely syntactic part that does not require semantic notions of truth and falsehood.

If we were to take away anything of philosophical import from Godel, it would be that we should be using either a paraconsistent logic (where statements can be both true and false without explosion) or an intuitionist logic (where statements can be neither true nor false). — Pfhorrest

Paraconsistency is a way out of incompleteness, but not on account of considerations of truth and falsehood but because contradictions are allowed. Again, Godel's theorem has a purely syntactical aspect as well as its semantical implications too.

Intuitionist formal logic is a proper subset of classical formal logic. Intuitionist logic is not a way out of incompleteness.

visual comments and props are the requirements of the media. — Wayfarer

Visual gimmicks and props are not required. One can give a talk orally and with supporting text and/or non-gimmicky visuals.

And I don't even object to visuals and props, except my point is that the ones in that video are stupid. The video is a collection of baubles.

And the video is a shallow attempt at entertainment while being not very informative, not clear even as a simplification, and egregiously misleading at certain points.

An example of the stupidity is spending time on Godel's inanition. It has nothing to do with the subject. And the video includes several seconds holding on a cartoon representation of a plate of food, suggesting that is what Godel passed up in refusing to eat. As if we need to be shown what a plate of food looks like. How childishly stupid.

Gödel offered a proof that math is either inconsistent or incomplete and that the dilemma is undecidable. — Gregory

That's not Godel's theorem. You don't know what Godel's theorem is.

Previously asked:

Gödel was trying to find a way to make a line in between what can be known and what can not
— Gregory

Where did you read that? — TonesInDeepFreeze

It oversimplifies to the point of being terribly misleading. One glaring mistake is not recognizing that undecidability follows immediately from incompleteness.

And the visual gimmicks and props are not helpful.

Gödel was trying to find a way to make a line in between what can be known and what can not — Gregory

Where did you read that?

What I've said only would make sense to someone who has thought spiritually — Gregory

So, according to you, a necessary condition for making sense of your idea is thinking spiritually. But meanwhile a necessary condition for making sense of Godel's theorem is knowing what it is.

if you don't like the idea of God at least be up to saying so — Gregory

What I don't like is people spouting about Godel's theorem without knowing what it is. If you're not up to finding out what Godel's theorem is then at least be up to saying so.

Yes, nice article.

I should revise what I said. Maybe something like this (not necessarily in this order):

(1) Creating new systems.

(2) Ingeniously proving theorems.

(3) Proving theorems in a way that engenders new techniques (such as Cohen's forcing).

(4) Posing questions or conjectures.

(5) Critiquing (such as Brouwer's prosecution against classical mathematics).

(6) Thinking up good pedagogical explanations by which people can better understand mathematics.

(7) Writing scorching ripostes to cranks. (Just kidding about that one.)

The latter. I guarantee it.

If we can know God perfectly, we can prove everything in mathematics once we fully know him and Godel's theorem will not apply. — Gregory

From that it is apparent that you don't know what Godel's theorem is. Your commentary is relevant to what you think Godel's theorem is but not relevant to Godel's theorem.

Russel — Metaphysician Undercover

You wrote 'Russel' twice. It's 'Russell'.

Tracking recent points Metaphysician Undercover has either evaded or failed to recognize that he was mistaken.

As previous:

https://thephilosophyforum.com/discussion/comment/544630

And now:

we ought not say that the numeral 2 says the same thing as the Hebrew symbol.
— Metaphysician Undercover

We sure better say that '2' and 'bet' name the same number. Otherwise, translation would be impossible. If '2' and 'bet' named different numbers then English speakers and Hebrew speakers could never agree on such ordinary observations as that the quantity (you like the word 'quantity') of apples in the bag is the same whether you say it in English or in Hebrew. — TonesInDeepFreeze

This would be very clear to you if you would consider all the different numbering systems discussed on this forum, natural, rational, real, cardinals, ordinals, etc..
— Metaphysician Undercover

Ah, red herring.

The point is whether the English numeral and the Hebrew numeral name the same number. That is unproblematic. It is not a contradiction or illogical for an object to have different words denoting it.

It is an unrelated point that there are different kinds of numbers. — TonesInDeepFreeze

The same symbol has a different meaning depending on the system. If we do not keep these distinguished, and adhere to the rules of the specific system, we have equivocation.
— Metaphysician Undercover

You have it reversed, as you often do.

Yes, by making clear that certain symbols are used differently in different contexts, we avoid equivocation. Using a symbol in more than one way is one-to-many: one (one symbol) to many (many different meanings). And one-to-many is a problem if we don't make clear contexts.

But with the English numeral and Hebrew numeral, we're not talking about one-to-many. Rather, we are talking many-to-one: many (two symbols) to one (one number).

Either you are actually so confused that you can't help but reversing or you are dishonest trying to make the reversal work for you as an argument. I'm guessing the former, since, even though you are often dishonest, more often it is apparent that you are just pathetically confused. — TonesInDeepFreeze

.That means for you to state which dots come before other dots, for each dot.
— TonesInDeepFreeze

Order is not necessarily temporal
— Metaphysician Undercover

YOU were the one harping on temporality and saying that things were place in order temporally by people. I don't rely on temporality. I didn't say that 'before' is 'before' only in a temporal sense. — TonesInDeepFreeze

Why must the symbol "2" represent a mathematical object, the number two, and the number two represents a quantity of two individuals? We don't say that the word "tree" represents a conceptual object, tree, and this concept represents the individual tree.
— Metaphysician Undercover

Because 'tree' is not a proper noun. — TonesInDeepFreeze

There are many mathematicians to whom truth and falsity are very relevant. And not just model-theoretic truth and falsity.
— TonesInDeepFreeze

Yes, but in what sense? — fishfry

In a realist sense, whatever the mathematician's or philosopher's concept of mathematical realism. In particular, many mathematicians believe that the continuum hypothesis is true or false, in a real sense.

From the very first paragraph of the introduction to Kunen's Set Theory: An Introduction to Independence Proofs, he says: "All mathematical concepts are defined in terms of the primitive notions of set and membership." — fishfry

Yes, that is quite common. However, as I said, in a strict technical sense, we don't need to regard 'set' as primitive. 'set' does not occur in the axioms, and is not even a primitive in the language.

I think it's fair to say that set is an undefined term — fishfry

I can give you a definition using only 'member of'.

You, and Tones alike (please excuse me Tones, but I love to mention you, and see your response. Still counting?), are simply in denial of these logical fallacies existing in the fundamental principles of mathematics, and you say truth and falsity is irrelevant to the pure mathematicians. — Metaphysician Undercover

It is, at best, ambiguous whether the third 'you' applies only to fishfry or "fishry and Tones alike". If you meant to be clear, then you would have been clear which part of your claim is meant for only fishfry and and which for both of us "alike".

And why, other for the purpose of trolling (as you admit to mentioning me only to provoke response) drag me into your scattershot claim?

Moreover, you have not shown that a logical fallacy, let alone one that I have denied. And, you skip my rebuttal to you regarding fallacies.

I am finding that to be the best tactic in dealing with the type of nonsense you throw at me. — Metaphysician Undercover

You evade even the most clear refutations.

What is "THE INHERENT" order you claim that the dots have?
— TonesInDeepFreeze

The one in the diagram. Take a look at it yourself, and see it. — Metaphysician Undercover

You repeat your evasion yet again.

The set of points has many different orderings. No one ordering is privileged as "the inherent ordering".

If there is a certain ordering that you think is "the inherent ordering" then tell us what it is. Point to each dot and tell us which dots it comes before and which dots it comes after. That is what is meant by an ordering in this discussion (a total linear ordering).

I have a jar of sand in front of me. There are many orderings of the grains, but there is no single particular ordering that is "THE inherent ordering".

But you will evade that point yet again.

I have no problem with what people say in everyday language, about the number of students in the class, the number of chairs in the room, the number of trees in the forest, etc., where I have the problem is with what mathematicians say about numbers alone, without referring to "the number of ..." — Metaphysician Undercover

You are welcome to change your position, but it's not what it was:

The number does not represent how many individuals there are.

The number is how many individuals there are.
— Luke

Well no, this is not true.
— Metaphysician Undercover — TonesInDeepFreeze

In everyday language, the number is how many individuals there are.

As you describe sets, order is an attribute, or property of the set — Metaphysician Undercover

I didn't say that.

How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction? — Metaphysician Undercover

First, aside from answering you, to make clear, there is no "same time" term in set theory.

There is no contradiction in the existence of more than one ordering of a set. Example:

{1 2} has two linear orderings:

{<1 2>}

and

{<2 1>}

Or, using concretes:

{Bob, Sue} has two linear orderings:

(<Bob Sue>}

and

{Sue Bob}

See, I answer your questions. Howzabout you answer mine?:

Do you think music theory is wrong like mathematics? And, if so, what are the bad things things that happen from music theory? And do you know any music theory?

Between you and fishfry, the two of you do not even seem to be in agreement as to whether a set has order or not. — Metaphysician Undercover

I speak for myself and not for fishfry.

"Set has order" is not a rubric I would use except loosely.

I've stated exactly the case. Refer to what I have said, not tangling it up with another poster.

For every set S with cardinality greater than 1, there exists more than one total linear ordering of S.

Fishfry resolves this by saying that a set has no order, so order is not a property of a set. But then it appears like fishfry wants to smuggle order in with some notion of possible orders. — Metaphysician Undercover

Whether or not that is a fair characterization of fishfry's view, it does not bear on my comments; I speak only for myself. I will say that "possible" is sometimes used informally in mathematics, but in a context such as this, we can dispense with 'possible'. We prove existence statements about sets and orderings of the sets. We don't have to say they are "possible" orderings.

. You say that not only does a set have order, but it has a multitude of different orders at the same time. See what happens when you employ contradictory axioms? Total confusion. — Metaphysician Undercover

Wrong. You have not shown any contradiction in the axioms. People take liberties from the strict formulations and discuss set theory in informal ways that may be misunderstand by an ignorant, stubborn and confused crank. But a reasonable person informed in the subject would very well understand that phrases such as "lack of inherent order" or "possible other orders" can be resolved to instead more definite formalisms without loose terminology such as "inherent" or "possible".

I'll give it to you again:

If S has cardinality greater than 1, then there are more than 1 total linear orderings of S.

you [Metaphysician Undercover] are wrong in believing that anyone is claiming that math is stating metaphysical truth — fishfry

There are mathematicians and philosophers who do claim that mathematics states metaphysical (platonic, or however it may be couched) truths.

But truth and falsity ARE irrelevant to pure mathematicians. — fishfry

There are many mathematicians to whom truth and falsity are very relevant. And not just model-theoretic truth and falsity.

my diagram was intended to help make a point, but it clearly didn't work very well — fishfry

It didn't work to bring Metaphysician Undercover to reason. But it was a fine illustration for anyone with the ability and willingness to comprehend.

Set is an undefined term, just as point and line are undefined terms in Euclidean geometry. — fishfry

Not quite. The only primitive of set theory is 'element of'. We don't need 'set' for set theory as we need 'point' and 'line' for Euclidean geometry.

Of course, in our background understanding we also take the notion of a 'set' as a given. But in actual formality, 'set' can be defined from 'element of'.

The ZF axioms fully characterize what sets are, by specifying how sets behave. — fishfry

But there are important properties of sets that are not settled by the axioms, so many set theorists do not believe that the axioms fully characterize the sets.

As to what sets actually are, nobody has the slightest idea. — fishfry

I have an exact idea, relative to the the undefined 'element of'. For me, 'set' is not the notion itself of which I could not explicate, but rather the actual primitive 'element of'.

Isn't "important aspect" weaselly enough? — fishfry

I said 'important aspect'. I don't see anything "weaselly" about that.

I didn't say "all" or "most," just an important aspect. — fishfry

What you said is:

The essence of creativity in math is to make up new rules. — fishfry

I didn't claim that you said "all" or "most". Rather, I shared my impression that most mathematical creativity is in theorem proving. I don't take either one of devising new systems or theorem proving to be the essence of mathematical creativity, but would be happy to agree that together they combine to make the essence of mathematical creativity.

logical fallacies existing in the fundamental principles of mathematics — Metaphysician Undercover

A logical fallacy is an improper argument form. You've not shown any fallacy in mathematics. Of course, you may reject the axioms, and you may reject the rules of inference and claim that the rules are not proper. But you've not shown any argument why we should consider the rules improper.

Moreover, the consistency of the rules is proven finitistically. (I forgot, but I think that finitistically proves the soundness of the rules too.) That is how intellectually thorough and honest mathematics is. Mathematics even proves its own proof methods are consistent and sound, and does it finitistically. And mathematical logic even investigates alternate proof methods that incorporate alternative views of mathematics and logic such as constructivism, paraconsistency, relevance logic, etc.

Said yet another way: Saying (1) "there is not one particular ordering that is 'the inherent ordering'" is not saying (2) "there is no ordering".

The essence of creativity in math is to make up new rules. — fishfry

Devising new frameworks and systems is an important aspect of creativity in mathematics. But, while I can't properly quantify, it seems to me that most of mathematical creativity is in proving theorems.

as I understand it, the point iof the diagram of "dots" is that the elements of the set have no inherent numerical order or sequence. — Luke

I don't speak for the originator of the illustration, but I take it to mean that there is not an "inherent" order (there is not one particular ordering that is the "inherent ordering"), whether numerical or of any kind.

Metaphysician Undercover with the same dishonest claim:

Contradiction may be implied. Here's Wikipedia's opening statement:
'In traditional logic, a contradiction consists of a logical incompatibility or incongruity between two or more propositions."

The problem is that you refuse to recognize that an arrangement of points on a plane, logically implies order, therefore "an arrangement of points on a plane without order" is contradictory. — Metaphysician Undercover

(1) Again, I have said at least a few times already that sets have orderings. Sets of cardinality greater than 1 have more than one ordering.

You even put in QUOTE MARKS "an arrangement of points on a plane without order", which is something I never said. You are lying about me.

(2) A set of statements is inconsistent if and only if it implies a contradiction. A contradiction is a statement and its negation.

You claim that I have advanced a contradiction. So, you should be able to show that anything I've said implies a contradiction. But these two statements are not a contradiction:

* For every set, there are orderings of the set.

* For sets of cardinality greater than 1, there is no single ordering that is "THE INHERENT ORDERING".

(I'll add that if one want to define 'the inherent ordering' for certain sets, such as the ordering by membership for ordinals, or the standard ordering of the reals, etc., then that's okay with me. But the point is that there is not such a definition for sets in general.)

Now more of the ignorance and confusion of Metaphysician Undercover:

If one could predict the bad things that were going to happen, before they happened, then we could take the necessary measures to ensure that they don't happen. — Metaphysician Undercover

I'm not asking you what particular bad things you think will happen, but what kind of bad things. At least you could say what is the general nature of the bad things you think will happen (you do below, I'll get to it).

It's like asking me what accident are you going to have today. — Metaphysician Undercover

No it's not. That's a strawman argument by you. I'm not asking you to predict that Joe Blow in Paducah will burn his toast tomorrow. Obviously, that would be ridiculous. So obviously it's not what I'm asking. I'm asking what is the general category of bad things you are warning against (you do below, I'll get to it). .

The biggest problem, I think, is the complete denial of the faults, from people like you. — Metaphysician Undercover

I mentioned in a post above that I don't have such a denial. But at least you do mention a general kind of bad effect you have in mind.

(1) The most present example is your denial of the utter ludicrousness of you your ignorant, confused, illogical, and deceptive ideas about mathematics. No matter how clearly those are pointed out to you, you evade and deny.

(2) Any subject can have people in denial about its faults. If we deleted intellectual work on the basis that there are certain people that have too rigid adherence, then we'd have virtually no intellectual work to refer to for all human history.

(3) So when you said "bad things", it turns out that for the most part, these bad things are that people explain to you how mathematics actually works so that they may disabuse you of your ignorant and confused imaginings about it.

This creates a false sense of certainty. That's why it's like religion, you completely submit to the power of the mathematics, with your faith, believing that your omnibenevolent "God", the mathematics would never mislead you. — Metaphysician Undercover

I have addressed the "religion" claim in detail in other posts. You ignore what I said. That is your favorite argument tactic: Don't recognize the points others make and instead just keep repeating your false and confused claims.

Also, you are virtually lying about me again by claiming that I believe that mathematics is an omnibevolent "God" that would never be misleading.

we ought not say that the numeral 2 says the same thing as the Hebrew symbol. — Metaphysician Undercover

We sure better say that '2' and 'bet' name the same number. Otherwise, translation would be impossible. If 2' and 'bet' named different numbers then English speakers and Hebrew speakers could never agree on such ordinary observations as that the quantity (you like the word 'quantity') of apples in the bag is the same whether you say it in English or in Hebrew.

This would be very clear to you if you would consider all the different numbering systems discussed on this forum, natural, rational, real, cardinals, ordinals, etc.. — Metaphysician Undercover

Ah, red herring.

The point is whether the English numeral and the Hebrew numeral name the same number. That is unproblematic. It is not a contradiction or illogical for an object to have different words denoting it.

It is an unrelated point that there are different kinds of numbers.

(By the way, for naturals, ordinals and cardinals, they are the same.)

The same symbol has a different meaning depending on the system. If we do not keep these distinguished, and adhere to the rules of the specific system, we have equivocation. — Metaphysician Undercover

You have it reversed, as you often do.

Yes, by making clear that certain symbols are used differently in different contexts, we avoid equivocation. Using a symbol in more than one way is one-to-many: one (one symbol) to many (many different meanings). And one-to-many is a problem if we don't make clear contexts.

But with the English numeral and Hebrew numeral, we're not talking about one-to-many. Rather, we are talking many-to-one: many (two symbols) to one (one number).

Either you are actually so confused that you can't help but reversing or you are dishonest trying to make the reversal work for you as an argument. I'm guessing the former, since, even though you are often dishonest, more often it is apparent that you are just pathetically confused.

.That means for you to state which dots come before other dots, for each dot.
— TonesInDeepFreeze

Order is not necessarily temporal — Metaphysician Undercover

YOU were the one harping on temporality and saying that things were place in order temporally by people. I don't rely on temporality. I didn't say that 'before' is 'before' only in a temporal sense.

Still you are evading the challenge: What is "THE INHERENT" order you claim that the dots have?

Whatever you like - temporal or not - you claim that sets have "AN INHERENT" order. So what is the inherent order of those dots?

So if you cannot see order in an arrangement on a two dimensional plane, I don't see any point in discussing "order" with you. — Metaphysician Undercover

STOP

Stop driving right past the point I have told you over and over.

Just read this. Take a moment. And try to understand:

I have said several times that there ARE orderings. There are MANY orderings. So there is not a single ordering that can be called "THE INHERENT" ordering.

Indeed, for a finite set of cardinality n, the number of (total linear) orderings of is n factorial ('n!' in math notation). And when n>1, n!>1, so there are MORE THAN ONE orderings of the set.

And stop trying to make it seem that I deny that there are no orderings of a set.

What I am asking is why can't the symbol "2" be used to represent a quantity of two individuals, — Metaphysician Undercover

It is used to denote the quantity two.

Why must the symbol "2" represent a mathematical object, the number two, and the number two represents a quantity of two individuals? We don't say that the word "tree" represents a conceptual object, tree, and this concept represents the individual tree. — Metaphysician Undercover

Because 'tree' is not a proper noun.

In reality we simply use the word "tree" to represent a tree, and we use the symbol "2" to represent a quantity of two individuals. — Metaphysician Undercover

The number does not represent how many individuals there are.

The number is how many individuals there are.
— Luke

Well no, this is not true. — Metaphysician Undercover

Well yes, it is true.

Start with what people say in everyday language. Jack says, "What is the number of students in the class?" and Sue says, "The number of students in the class is two".

The number IS the number of students. That is everyday language.

And mathematics captures that thinking.

You want for us to regard everyday language working differently, working according to your own nut-case confusions.

Next, a break to repeat a question:

The part that doesn't make sense is when you move deeper into the theory. This is just like mathematics.
— Metaphysician Undercover

I want to understand: Are you saying that music theory is wrong just as you say mathematics is wrong? And, by the way, do you know any music theory? — TonesInDeepFreeze

Next, Metaphysician Undercover's trolling:

I love to mention you, and see your response. — Metaphysician Undercover

I pointed out that continually you mention me without stating the context or quotes, thus making it seem that I have played a certain role or taken a certain position in an unspecified exchange with you. And above you admit that you do this to provoke my response regarding that. That is the very definition of 'trolling'. And you admit it. You're an obnoxious bane.

We still have this list of corrections, challenges, and questions that Metaphysician Undercover has not answered: https://thephilosophyforum.com/discussion/comment/544630

And now we have another installment of ignorance, confusion, illogic, dishonesty, and trolling from him.

Let's start with the dishonesty:

You [fishfry], and Tones alike [...], are simply in denial of these logical fallacies existing in the fundamental principles of mathematics, and you say truth and falsity is irrelevant to the pure mathematicians. — Metaphysician Undercover

I don't speak for fishfry, but the second 'you' above appears to include both of us. But I have never said, implied, or remotely suggested, that truth and falsity are irrelevant to pure mathematics. So you are lying to suggest that I did.

the complete denial of the faults, from people like you. — Metaphysician Undercover

I have never claimed that mathematics, or classical mathematics, is exempt from criticisms. Indeed I have said at least a few times that I am interested in discussion of criticisms, including from such tenets as predicativism, constructivism, finitism, formalism, relevance logic, and even paraconsistent logic. Also, the question of infinite regress in meta-theory. Also, objections that set theory is too rococo and overshoots the target of mathematics for the sciences. Also, reverse mathematics. And I have not opined that all these critiques are incorrect and especially I have not opined that they are not worthwhile. So you are lying about me.

Do you recognize that the word 'tree' is not a tree?

That the word 'Chicago' is not the city of Chicago?

That the word 'courageousness' is not the atrribute courageousness?

That the word 'Ahab' is not a fictional character?

That the word 'liberty' is not liberty itself nor the concept of liberty?

Yes?

But you fail to recognize that the word 'two' or the symbol '2' are not the number 2.

Reminding again, more of the recent points you have failed on:

What else could demonstrate falsity other than a reference to some form of inconsistency?.
— Metaphysician Undercover

Falsity is semantic; inconsistency is syntactical.

Given a model M of a theory T, a sentence may be false in M but not inconsistent with T.
— TonesInDeepFreeze — TonesInDeepFreeze

An axiom is expressed as a bunch of symbols, so it must be interpreted.
— Metaphysician Undercover

Formulas don't have to be interpreted, though usually they are when they are substantively motivated.
— TonesInDeepFreeze — TonesInDeepFreeze

If in interpretation, there is a contradiction with another principle then one or both must be false.
— Metaphysician Undercover

It might not be a matter of principles but of framework. Frameworks don't have to be evaluated as true or false, but may be regarded by their uselfulness in providing a conceptual context or their productivity in other ways.
— TonesInDeepFreeze — TonesInDeepFreeze

Notice there is an exchange of "equal" and "same"
— Metaphysician Undercover

Even though there is nothing wrong with taking 'equal' to mean 'same', the axiom of extensionality doesn't require such mention.

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

"=' is mentioned, but not "same".
— TonesInDeepFreeze — TonesInDeepFreeze

/

And new ones:

What is added or multiplied is the quantity or number of individuals. The number is of the individual, a predication, and what is added or subtracted is the individuals, not the number.
— Metaphysician Undercover

That's just a plain contradiction from one sentence to the next. — Luke

That people vehemently support and defend fundamental axioms which may or may not be true, refusing to analyze and understand the meaning of these axioms, simply accepting them on faith
— Metaphysician Undercover

But in the philosophy of mathematics, which includes many mathematicians themselves, people do investigate, question, and debate the axioms - giving reasoned arguments for and against axioms. It's just that you are ignorant of that. — TonesInDeepFreeze

The part that doesn't make sense is when you move deeper into the theory. This is just like mathematics.
— Metaphysician Undercover

I want to understand: Are you saying that music theory is wrong just as you say mathematics is wrong? And, by the way, do you know any music theory? — TonesInDeepFreeze

This is why mathematics really is like religion. We are required just to accept the rules, on faith, follow and obey, without any real understanding.
— Metaphysician Undercover

That is false. It's the opposite. That describes the grade school memorization and regurgitation of tables and rules for basic addition, subtraction, multiplication, and division that you find so suitable. Mathematics though provides understanding of the bases for those rules. — TonesInDeepFreeze

without any order
— Metaphysician Undercover

You are obfuscating by sliding between adressing "order" and "actual order" (or "inherent order"). That's typical of your intellectual sloppiness.

It is not the case that there are not orderings. The point though is that there is not a single ordering that is "THE actual ordering". There are many orderings and they are actual even though 'actual' is gratuitious. — TonesInDeepFreeze

/

Also, you continue to mention me sometimes without quote or context, placing me in certain roles in the dialectic that I have not taken.

First thing I really want to know what are the bad things that you think mathematicians and scientists are going to cause to happen? What bad things do you think are going to happen if mathematicians contnue to regard numerals and numbers as not the same, as mathematicians have for a very long time? Why haven't these bad things already happened? What do you think are the bad things that are going to happen if mathematicians continue to recognize that sets have many different orderings that there is no "THE actual ordering"?

There is no need to assume that the number 2 is distinct from the symbol, to do basic arithmetic.. — Metaphysician Undercover

Suppose the number 2 is not distinct from the numeral '2'. Suppose also that the number 2 is not distinct from the Hebrew numeral for 2. Then both the numeral '2' and the Hebrew numeral for 2 are the same. But they are not.

The numerals are not the numbers. If they were, then anybody who used different numerals would be naming different numbers. But they're not. Everybody is naming the same number 2 whether they use the numeral '2' or the Hebrew numeral of the Roman numeral or the word 'two' or any of many other names for the number 2. A child can understand that.

Why not just say that the symbols "'1" and "2" represent how many individuals there are, directly? — Metaphysician Undercover

We can, and we do. But also we wish to mention in particular that the number of individuals is 1 or 2 as may be the case.

Fishfry posted the order, it's right here:
↪fishfry — Metaphysician Undercover

You are totally confused. That's a picture of dots in a disk. It's not an ordering.

What more do you want? — Metaphysician Undercover

For you to state what you claim to be the "actual ordering of the dots" - as I asked about five times already. And to give reason why that is the "actual ordering" as opposed to other orderings. That means for you to state which dots come before other dots, for each dot. And then say why that ordering you chose is "actual" while other orderings we can choose are "not actual".

How can you not see that 'points in a plane without a particular ordering' is a blatant contradiction? — Metaphysician Undercover

A contradiction is a statement and its negation.

One things virtually all cranks have in common is claiming to point out a contradiction when all they're doing is pointing out something that they happen to disagree with. There should be a name for that fallacy.

Anyway, you cannot see that there are many orderings of that finite set of points, but no one of those many ordefings is "THE actual ordering". They ALL are actual orderings. And "actual" is gratuitious anyway.

One more time: There are many different orderings. But not one of them is privileged as being more "actual" or "inherent" than the others. Do really still not understand that?

The rest of your post is just different ways of you repeating your misunderstanding sourced in your not knowing what an ordering is.

Again, you use the word 'ordering' or 'order' in a way that is neither their use in mathematics nor even in everday speech. You assert an entirely personal notion and usage. And your own usage is not even coherent onto itself nor consistently applied by you. Yet you expect everyone else to come around to adopt your personal usage and then also to revise their clear and common notions about basic mathematics to conform to your ignorant, uneducated, and incoherent concept of mathematics. Classic crank to the core.