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.

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".

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.

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. — TonesInDeepFreeze

Isn't "important aspect" weaselly enough? I didn't say "all" or "most," just an important aspect. A lot of the big breakthroughs do involve radically new ways of seeing things. I'm not quantifying it, was just pointing out to @Metaphysician Undercover that there's a lot more to math than just following arbitrary rules.

Don't you see that I said math is not like chess. Therefore I do not treat math like chess. I answered your question. — Metaphysician Undercover

You seem to want to place math on some kind of pedestal, as if someone is claiming it's absolute truth, then you point out that it's not absolute truth. But that's a classic strawman. Nobody is claiming math is absolute truth but you. Math is just a way that we formalize certain intuitive notions. But it's a formalization and not intended to be the "real thing." So your metaphysical points may be right, but you are wrong in believing that anyone is claiming that math is stating metaphysical truths. On the contrary, all mathematical truths are relative. IF this THEN that. Math takes no position on whether "this" is true or even meaningful. Math only says that if you accept this then you can prove that. You are the one trying to make more of this than anyone intends. Can you see that?

Then your complaint is with the physicists, engineers, and others; and not the mathematicians, who frankly are harmless.
— fishfry

Obviously not, as you've already noticed, — Metaphysician Undercover

"Mathematics considered harmful," LOL. The reference is to the early days of programming, when GOTO statements were prevalent and led to messy, "spaghetti" code. Computer scientists Edsgar Dijkstra published an article called Go To Statement Considered Harmful. Ever since then, "X considered harmful" is an inside joke in CS. This is the first time I've ever seen anyone claim that mathematics is considered harmful! Hence my amusement. Perhaps you can 'splain yourself.

No, my complaint is with the fundamental principles of mathematicians, As explained already to you, violation of the law of identity, contradiction, and falsity. — Metaphysician Undercover

Nonsense, nonsense, and nonsense. You haven't made your case in three years of trying.

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

But truth and falsity ARE irrelevant to pure mathematicians. Russell famously quipped, "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." Don't you think he was recognizing and responding to exactly the point you are making? What do you make of old Bertie's remark?

In case you forgot, you posted a diagram with dots, intended to represent a plane with an arrangement of points without any order. This is what I argued is contradictory, "an arrangement... without order". — Metaphysician Undercover

And as I already said, my diagram was intended to help make a point, but it clearly didn't work very well, so forget it. Sets are defined by the rules of set theory, nothing more and nothing less. And once again I pointed you to the axiom of extensionality. So nevermind the dots in the circle. If the analogy was lost on you, then forget it. You can't still be arguing against an example. Try understanding the axiom of extensionality.

And this was representative of our disagreement about the ordering of sets. You insisted that it is possible to have a set in which the elements have no order. You implied that there was some special, magical act of "collection" by which the elements could be collected together, and exist without any order. — Metaphysician Undercover

No more magic that the rules of chess are magic when they say how the various pieces move. The rules of set theory say how sets behave. The analogy is perfect, whether you get it or not.

What you are in denial of, is that if the elements exist, in any way, shape, or form, then they necessarily have order, because that's what existence is, to be endowed with some type of order. — Metaphysician Undercover

Existence is to be endowed with order? Now that is utter nonsense. One example that comes to mind is the famous counterexample to the identity of indiscernibles, in which we posit a universe consisting of two identical spheres. You can't distinguish one from the other by any property, even though the spheres are distinct. This example also serves as a pair of objects without any inherent order.

But "existence is to be endowed with order?" Man you are just flailing, making things up instead of honestly engaging.

You tell me, just imagine a plane, with points on the plane, without any order, and I tell you I can't imagine such a thing because it's clearly contradictory. If the points are on the plane, then they have order. And you just want to pretend that it has been imagined and proceed into your smoke and mirrors tricks of the mathemajicians. I'm sorry, but I refuse to follow such sophistry. — Metaphysician Undercover

Well ok. I can't add much. As I've noted, we're kind of done here. I have nothing to add and you aren't interested in engaging with math. You prefer to reject it wholesale and that's your privilege. I can't do anything about it. Not for lack of trying.

Why not give it a try? I can argue with the fantasies in your head, demonstrating that they are contradictory. So please explain to me how you think you can have a collection of elements, points, or anything, and that collection has no order. Take this fantasy out of your head and demonstrate the reality of it. — Metaphysician Undercover

Reality? I make no claims of reality of math. I've said that a hundred times. YOU are the one who claims math is real then complains that this can't possibly true. The solution is that nobody claims math is "real" in the sense that you mean. Of course math is real in terms of implications: if this, then that. But math makes no claims as to the truth of "this." Only IF this THEN that. Nobody says sets are "real" in any meaningful sense. Sets are abstract thingies (you don't like the word objects) that behave according to rules; just like chess pieces. You don't want to get that but your refusal to get that is the root cause of your misunderstandings. Nobody claims math is real in the sense that YOU use the word real.

But on the other hand math IS real, in the sense that, for example, the rules of group theory fully encapsulate the behavior of reversible transformations, like addition and subtraction, or multiplication and division, or rotating counterclockwise by 47 degrees and rotating clockwise by 47 degrees. The math of group theory is abstract and not "real" as YOU use the word real, but the relationships encoded by group theory ARE real. And you simply won't grapple with this. I don't know why.

The dots. I believe, were supposed to be a representation of points on a plane. The points on a plane, I believe, were supposed to be a representation of elements in a set. And you were using these representations in an attempt to show me that there is no inherent order within a set. So, are you ready to give it another try? Demonstrate to me how there could be a set with elements, and no order to these elements. — Metaphysician Undercover

https://en.wikipedia.org/wiki/Axiom_of_extensionality
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

That's all there is too it, since you take pot shots rather than attempt to derive understanding from visual analogies. The ZF axioms fully characterize what sets are, by specifying how sets behave. As to what sets actually are, nobody has the slightest idea. Set is an undefined term, just as point and line are undefined terms in Euclidean geometry.

I've explained to you the problem. You describe the set as a sort of unity. And you want to say that the parts which compose this unity have no inherent order. Do you recognize that to be a unity, the parts must be ordered? There is no unity in disordered parts. Or are you going to continue with your denial and refusal to recognize the fundamental flaws of set theory? — Metaphysician Undercover

No. I describe a set as whatever obeys the ZF axioms. There are alternative axiom systems that describe a different notion of set. We do have intuitive, casual, naive examples of what we mean, but presenting these to you (collections of things, bags of groceries, circles containing red dots) are lost on you and only give you ammunition for childish objections.

So a set is whatever ZF says a set is. Just like the knight moves the way the rules of chess say the knight moves. They are both formal systems with no referent in the real world.

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.

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'.

Hopefully others will correct me if I'm wrong but, 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. Otherwise, you should be able to number the elements from 1 to n and explain why that is their inherent order. — Luke

Actually "numerical order" (whatever that is supposed to mean in reference to a diagram of dots) was not specified. It was simply asserted that the elements have no inherent order.

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. — TonesInDeepFreeze

Are you taking lessons from Luke on how to make strawman interpretations? Read the quoted passage. Fishfry and Tones are in denial of the logical fallacies, and "you" (directed at fishfry only) talk about truth and falsity not being relevant to pure mathematics. How is that sentence so difficult for you to read properly?

You ignore what I said. That is your favorite argument tactic: — TonesInDeepFreeze

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

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.

Start with what people say in everyday language. — TonesInDeepFreeze

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 ..."

Sets of cardinality greater than 1 have more than one ordering. — TonesInDeepFreeze

Look at it this way Tones. As you describe sets, order is an attribute, or property of the set. How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction? 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. However the set is already defined as not having the property of order, therefore order is impossible.

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. Fishfry says that a set has no inherent order. 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.

Nobody is claiming math is absolute truth but you. — fishfry

Wow, that's an even worse interpretation of what I'm saying than TIDF's terrible interpretation. I argue to demonstrate untruths in math, and you say I'm claiming math is absolute truth. This thread has gone too far. I think you're cracking up.

Don't you think he was recognizing and responding to exactly the point you are making? — fishfry

No, I have read a fair bit of Russel and he was in no way responding to the same points I'm making. More precisely he was helping to establish the situation which I am so critical of. Remarks like that only inspire mathematicians to produce more nonsense.

Try understanding the axiom of extensionality. — fishfry

We've been through the axiom of extensionality you and I, in case you've forgotten. It's where you get the faulty idea that equal to, means the same as.

One example that comes to mind is the famous counterexample to the identity of indiscernibles, in which we posit a universe consisting of two identical spheres. You can't distinguish one from the other by any property, even though the spheres are distinct. This example also serves as a pair of objects without any inherent order. — fishfry

Contradiction again. If you cannot distinguish one from the other, you cannot say that there are two. To count two, you need to apprehend two distinct things. But to say that you cannot distinguish one from the other means that you cannot apprehend two distinct things. Therefore it is false to say that there are two. So you are just proposing a contradictory scenario, that there are two distinct spheres which cannot be distinguished as two distinct spheres (therefore they are not two distinct spheres), hoping that someone will fall for your contradiction. Obviously, if one cannot be distinguished from the other, they are simply two instances of the same sphere, and you cannot say that there are two. And to count one and the same sphere as two spheres is a false count.

Reality? I make no claims of reality of math. I've said that a hundred times. YOU are the one who claims math is real then complains that this can't possibly true. — fishfry

Coming from the Platonic realist who claims the reality of "mathematical objects".

But math makes no claims as to the truth of "this." — fishfry

Right, just like the statement from Russel. That's why there is a real need for metaphysicians to rid mathematics of falsity. The mathematicians obviously do not care about festering falsities.

.

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. — TonesInDeepFreeze

I think we can defer to Gowers's great essay, The Two Cultures of Math, which he identifies as theory builders and problem solvers.

https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

Gowers: "The “two cultures” I wish to discuss will be familiar to all professional mathematicians.Loosely speaking, I mean the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and understanding theories"

This is very well said. Of course the distinctions are not clean cut. And the following resonates:

Atiyah: "Some people may sit back and say, “I want to solve this problem” and they sit down and say, “How do I solve this problem?” I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. "

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.

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

Well I'm a formalist sometimes and a Platonist other times. I made the point earlier to @Metaphysician Undercover that while math isn't "true" in the sense he thinks it's supposed to be, on the other hand it DOES express certain relational or structural truths. For example group theory expresses everything we could ever know about invertible transformations. Group theory expresses truths about such things. Yet formally, groups are sets, and sets have an very tenuous claim on being real. So somehow, math is fiction yet expresses deep structural truths. A lot of philosophers have said clever things about this.

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? I like my example. I don't think sets are particularly real. I don't ever try to defend the reality of sets. I don't believe there is an empty set or a set containing the empty set and the set that contains the empty set. But groups are defined as particular types of sets, and group theory expresses deep truths about invertible transformations. Out of nonsense, we get sense.

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. — TonesInDeepFreeze

It was a fine example indeed! But @Meta cleverly turned it against me and made me regret mentioning it. He does that so well! :-)

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. — TonesInDeepFreeze

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." Then as the text evolves, he builds up the usual hierarchy of sets starting from the empty set, and never seems to say what a set is; except that a set is one of the things built up. That's what I know about it, but I'm not equipped to dispute the fine points. I think it's fair to say that set is an undefined term; but there's no actual axiom that says, "Set is an undefined term." The whole business is kind of vague, actually.

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'. — TonesInDeepFreeze

Yes ok, I think I agree with your point. The larger point that I made to @Meta still holds. Informally a set is a bag of groceries or a circle with some red dots in it; but formally, a set has no meaning of its own outside of its behaviors under the axioms.

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. — TonesInDeepFreeze

Yes of course. Skolem thought that his Lowenheim-Skolem theorem showed that the notion of set isn't nearly as definite as we think. And of course there are various different axiomatic systems such as Von Neumann–Bernays–Gödel set theory, or Morse-Kelley set theory. And each such system is incomplete, so there are always true statements about sets that we can't prove. So it's fair to say that nobody knows for sure what a set is. Even the greatest set theorists; especially the greatest set theorists.

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'. — TonesInDeepFreeze

I don't think I'm in a position to argue that point one way or the other. Not entirely sure I'm following.

Actually "numerical order" (whatever that is supposed to mean in reference to a diagram of dots) was not specified. It was simply asserted that the elements have no inherent order. — Metaphysician Undercover

If not specified, then at least strongly implied in the same post:

What is the inherent order of the points in this set? Can you see that the points are inherently disordered or unordered, and that we may impose order on them arbitrarily in many different ways? Pick one and call it the first. Pick another and call it the second. Etc. What's wrong with that? — fishfry

Did you not read this when you went on to argue that the diagram has the order it has?

Just another example of your wilful ignorance and dishonest argumentation.

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'.

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

Russel — Metaphysician Undercover

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

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. — TonesInDeepFreeze

Yes ok I think you're right about 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? 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. However the set is already defined as not having the property of order, therefore order is impossible. — Metaphysician Undercover

I wasn't born wearing a hat but I can go buy one. The idea that a thing can't gain stuff it didn't have before is false. But I've already shown you how we impose order on a set mathematically. We start with the bare (newborn if you like) set. Then we pair it with another, entirely different set that consists of all the ordered pairs of elements of the first set that define the desired order.

So we start with the unordered set {a,b,c}. And we PAIR it with a different set {(a,b), (a,c), (b,c)} that defines the order a < b < c.

Now note well please. I am not saying that this process corresponds to any aspect of reality. It plainly doesn't. It is rather the way we formalize the idea of ordering a set.

Perhaps a key distinction I haven't explicitly called attention to is this:

I am not talking about reality. I am talking about how we use math to MODEL certain aspects of reality . With that distinction, I believe all our problems are solved. Math is a tool kit for modeling things we may be interested in. It's not the things themselves.

Also: "How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction?" Like ordering schoolkids by height, or alphabetically by name. Two different ways to order the same set. I can't understand why this isn't obvious to you.

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. Fishfry says that a set has no inherent order. 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

A set has no inherent order. That's the axiom of extensionality. A set may be ordered in many different ways, by pairing it with another set consisting of the ordered pairs that defined the order we're interested in.

Wow, that's an even worse interpretation of what I'm saying than TIDF's terrible interpretation. I argue to demonstrate untruths in math, and you say I'm claiming math is absolute truth. This thread has gone too far. I think you're cracking up. — Metaphysician Undercover

If you wish to argue the untruth of math, you won't get an argument from me. Math isn't supposed to be about the truth of particular things. Math says IF this THEN that. Or it reveals structural truths, as in group theory. But the contradictions you think you see, are only contradictions between math and what you THINK math is. But if your ideas are wrong, your argument fails.

No, I have read a fair bit of Russel and he was in no way responding to the same points I'm making. More precisely he was helping to establish the situation which I am so critical of. Remarks like that only inspire mathematicians to produce more nonsense. — Metaphysician Undercover

Not for nothin' is category theory called abstract nonsense. Never let it be said that mathematicians don't have a self-deprecating and self-aware sense of humor. Something you might aspire to emulate.

We've been through the axiom of extensionality you and I, in case you've forgotten. It's where you get the faulty idea that equal to, means the same as. — Metaphysician Undercover

Well your objection to it is wrong. Two sets are the same set if they have the same element. They are not two identical copies of the set. They're the same set. I don't see what your objection is, nor did I follow your objection several years ago. Jeez has it been that long? LOL.

Contradiction again. If you cannot distinguish one from the other, you cannot say that there are two. To count two, you need to apprehend two distinct things. But to say that you cannot distinguish one from the other means that you cannot apprehend two distinct things. Therefore it is false to say that there are two. So you are just proposing a contradictory scenario, that there are two distinct spheres which cannot be distinguished as two distinct spheres (therefore they are not two distinct spheres), hoping that someone will fall for your contradiction. Obviously, if one cannot be distinguished from the other, they are simply two instances of the same sphere, and you cannot say that there are two. And to count one and the same sphere as two spheres is a false count. — Metaphysician Undercover

You can't visualize two identical spheres floating in an otherwise empty universe? How can you not be able to visualize that? I don't believe you.

Here, look.

Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provide a model in which there are two distinct (numerically nonidentical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects even though they have all their properties in common.

https://en.wikipedia.org/wiki/Identity_of_indiscernibles

I did not make this up. I'm not clever enough to have made it up. I read it somewhere. Now you've read it.

Coming from the Platonic realist who claims the reality of "mathematical objects". — Metaphysician Undercover

I claim their mathematical existence and not their physical existence. But I admit that I'm not sufficiently knowledgable in philosophy to verbalize these subtle distinctions. Does the number 5 exist? Yes, as an abstract object. What does that mean? I can't personally say, but plenty of smart people have made the attempt.

Right, just like the statement from Russel. That's why there is a real need for metaphysicians to rid mathematics of falsity. The mathematicians obviously do not care about festering falsities. — Metaphysician Undercover

No, on the contrary. Mathematicians love festering falsities. And of course even this point of view is historically contingent and relatively recent. In the time of Newton and Kant, Euclidean geometry was the true geometry of the world, and Euclidean math was true in an absolute, physical sense. Then Riemann and others showed that non-Euclidean geometry was at least logically consistent, so that math itself could not distinguish truth from falsity. Then Einstein (or more properly Minkowski and Einstein's buddy Marcel Grossman) realized that Riemann's crazy and difficult ideas were exactly what was needed to give general relativity a beautiful mathematical formalism. The resulting "loss of certainty" was documented in Morris Kline's book, Mathematics: The Loss of Certainty. Surely the title alone must tell you that mathematicians are not unaware of the issues you raise.

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.)

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.)

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.) — TonesInDeepFreeze

LOL. So I've heard. But the larger point remains, that in the past century math has experienced a loss of certainty.

deplorably tendentious hatchet job — TonesInDeepFreeze

You say that like it's a bad thing.

If not specified, then at least strongly implied in the same post: — Luke

I really don't see how the qualification "numerical" is relevant , or even meaningful in the context of dots on a plane. So I don't see why you think it was implied. Fishfry is not sloppy and would not have forgotten to mention a special type of order was meant when "no inherent order" was said numerous times.

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

Sorry "Bertie", as fishfry says.

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

I believe all the relevant points were addressed. You don't seem to know how to read very well.

I wasn't born wearing a hat but I can go buy one. The idea that a thing can't gain stuff it didn't have before is false. But I've already shown you how we impose order on a set mathematically. We start with the bare (newborn if you like) set. Then we pair it with another, entirely different set that consists of all the ordered pairs of elements of the first set that define the desired order. — fishfry

The problem is that your demonstration was unacceptable because you claimed to start with a set that had no order. A newborn is not a thing without order, so the newborn analogy doesn't help you.

So we start with the unordered set {a,b,c}. — fishfry

You are showing me an order, "a" is to the right of "b" which is to the right of "c". And even if you state that there is a set which consists of these three letters without any order, that would be unacceptable because it's impossible that three letters could exist without any order.

If you insist that it's not the letters you are talking about, but what the letters stand for, or symbolize, then I ask you what kind of things do these letters stand for, which allows them to be free from any order? To me, "a", "b", and "c" signify sounds. How can you have sounds without an order? Maybe have them all at the same time like a musical chord? No, that constitutes an order. Maybe suppose they are non-existent sounds? But then they are not sounds. So the result is contradiction.

I really do not understand, and need an explanation, if you think you understand how these things in the set can exist without any order. What do "a", "b", and "c" signify, if it's something which can exist without any order? Do you know of some type of magical "element" which has the quality of existing in a multitude without any order? I don't think so. I think it's just a ploy to avoid the fundamental laws of logic, just like your supposed "two spheres" which cannot be distinguished one from the other, because they are really just one sphere.

Also: "How is it possible that a set can be ordered in one way and in a contrary way at the same time, without contradiction?" Like ordering schoolkids by height, or alphabetically by name. Two different ways to order the same set. I can't understand why this isn't obvious to you. — fishfry

Those are different ways, but not contrary ways.

A set has no inherent order. That's the axiom of extensionality. — fishfry

Huh, all my research into the axiom of extensionality indicates that it is concerned with equality. I really don't see it mentioned anywhere that the axiom states that a set has no inherent order. Are you sure you interpret the axiom in the conventional way? I have no problem admitting that two equal things might consist of the same elements in different orders. We might say that they are equal on the basis of having the same elements, but then we cannot say that the two are the same set, because they have different orders to those elements, making them different sets, by that fact.

You can't visualize two identical spheres floating in an otherwise empty universe? How can you not be able to visualize that? I don't believe you. — fishfry

Are you serious? If I can imagine them as distinct things, I know that they cannot be identical. That's the law of identity, the uniqueness of an individual,. A fundamental law of logic which you clearly have no respect for.

I did not make this up. I'm not clever enough to have made it up. I read it somewhere. Now you've read it. — fishfry

The argument of Max Black fails because pi is irrational. There is no such thing as a perfectly symmetrical sphere. The irrationality of pi indicates that there cannot be a center point to a perfect circle. Therefore we cannot even imagine an ideal sphere, let alone two of them.

No, on the contrary. Mathematicians love festering falsities. And of course even this point of view is historically contingent and relatively recent. In the time of Newton and Kant, Euclidean geometry was the true geometry of the world, and Euclidean math was true in an absolute, physical sense. Then Riemann and others showed that non-Euclidean geometry was at least logically consistent, so that math itself could not distinguish truth from falsity. Then Einstein (or more properly Minkowski and Einstein's buddy Marcel Grossman) realized that Riemann's crazy and difficult ideas were exactly what was needed to give general relativity a beautiful mathematical formalism. The resulting "loss of certainty" was documented in Morris Kline's book, Mathematics: The Loss of Certainty. Surely the title alone must tell you that mathematicians are not unaware of the issues you raise. — fishfry

Yes, it seems like mathematics has really taken a turn for the worse. If you really believe that mathematicians are aware of this problem, why do you think they keep heading deeper and deeper in this direction of worse? I really don't think they are aware of the depth of the problem.

I really don't see how the qualification "numerical" is relevant , or even meaningful in the context of dots on a plane. So I don't see why you think it was implied. Fishfry is not sloppy and would not have forgotten to mention a special type of order was meant when "no inherent order" was said numerous times. — Metaphysician Undercover

Perhaps “numerical” wasn’t the right word. The context of the post and the preceding discussion indicates that a “before and after” ordinality was implied, of the sort Tones recently mentioned:

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). — TonesInDeepFreeze

Your assertion that the diagram has an inherent order which can be discerned simply by looking at the diagram does not specify what that order is.

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

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

nevermind posted in error

Yes, it seems like mathematics has really taken a turn for the worse. If you really believe that mathematicians are aware of this problem, why do you think they keep heading deeper and deeper in this direction of worse? I really don't think they are aware of the depth of the problem — Metaphysician Undercover

Yes, it certainly looks like that faulty brick in the foundations of math will surely cause the giant structure to collapse. I wish I had known this before becoming a mathematician. :cry:

erhaps “numerical” wasn’t the right word. The context of the post and the preceding discussion indicates that a “before and after” ordinality was implied, of the sort Tones recently mentioned: — Luke

Before and after, are temporal terms. Fishfry had rejected the notion that "order" is based in spatial-temporal relations, and wanted an order based in quantity. But a quantity based order is what produces the problem I first referred to. If "2" refers to a quantity of objects in the context of "order", then it does not refer to a single object, the number 2. In any case, what distinguishes one thing from another, allowing for individuals, and quantity itself is spatial relations. So we're back to spatial relations as the bases of order. It's very clear that the subject was order of any sort, when "no inherent order" was mentioned.

It was my suggestion that "order" is fundamentally temporal, but fishfry produced examples of an order based on a judgement of better and worse, to discount that theory. So we really haven't agreed on any specific type of order yet. This is probably because we haven't agreed on what type of existence the things which are supposed to have no inherent order, but are capable of being ordered, have.

Yes, it certainly looks like that faulty brick in the foundations of math will surely cause the giant structure to collapse. I wish I had known this before becoming a mathematician. :cry: — jgill

I don't foresee any imminent collapse, but the structure ought to be dismantled because it supports the faulty worldview which is prevalent today, which is a sort of scientism. Fishfry perceives that physics has reached a sort of dead end in its endeavours, but refuses to acknowledge that the dead end is brought about by the principles employed (mathematics included) rather than the unintelligibility of the world itself.

The use of "infinity" which is the topic of this thread (believe it or not) is a very good example. I apprehend, that at the base of the idea of infinity in natural numbers, is the desire, or intention to allow that numbers can be used to count anything. There will never be something which cannot be counted because the numbering system has been designed to allow that the numbers can always go higher. This gives the appearance that everything is measurable.

The only drawback is that this renders infinity itself, (the principle which provides this capacity, that everything is measurable) as immeasurable. So in reality everything is measurable except the principles we use to measure with. What this means is that to understand the nature of measurement and infinity, we must place these into a different category from the category of things which are measurable, and understand it on those terms, as immeasurable. If we attempt to bring infinity into the category of things which are measurable, as is the trend in modern mathematics, because we want mathematics to enable science to be applicable everything, even the thing which by its own design is immeasurable, then we introduce contradiction (the immeasurable is measurable) and therefore unintelligibility into our principles. That is where we are today, we have allowed unintelligibility to inhere within our principles of measurement. As fishfry said "math has experienced a loss of certainty".

Fishfry and I really agree that the big picture is very hazy (uncertain). But when it comes down to determining the specific points where the haziness arises from, fishfry refuses to follow. It's like seeing smoke on the horizon and wondering why it's there. But when I point to the fire, fishfry refuses to acknowledge a relationship between the fire and the smoke on the horizon. Maybe if fishfry would accept the possibility of a relationship, a closer look would reveal smoke rising from the fire.

It was my suggestion that "order" is fundamentally temporal — Metaphysician Undercover

How does this account for the order of rank of military officers, or of suits in a game of bridge? Or the order of values of playing cards in Blackjack? Or the order of the letters of the alphabet? Or monetary value?

Before and after, are temporal terms. — Metaphysician Undercover

'before' and 'after' are often in a temporal sense, but clearly not exclusively. Not in English. And surely not in math that doesn't mention temporality.