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I see that you are confused about the most basic aspects of mathematics, language and reasoning. On certain points, your understanding is not even at the level of a six year old child. I'm offering you help here, though I doubt you'll take it in.

Where is the result of the operation denoted? — Metaphysician Undercover

You just now quoted me with the answer to that question:

'2+1' denotes the result of the operation

already denoted — Metaphysician Undercover

In a rigorous context, things are denoted by the method of interpretation of a language. In the usual interpretation, by the recursive method, the denotation of '2+1' is determined from the denotations of '2', '1' and '+'.

Otherwise there would be absolutely no purpose to the "=" because "2+1" on its own does not say 3. — Metaphysician Undercover

'2+1' does not have '3' in it, but '2+1' and '3' name the same object. To express that '2+1' and '3' name the same object we write:

2+1 = 3

Or, put another way, to assert that 2+1 equals 3 ('equals' also said as 'is identical with', also said as 'is the same object as') we write:

2+1 = 3

Otherwise there would be absolutely no purpose to the "=" because everything which 2+1 equals would already be said simply by saying "2+1". — Metaphysician Undercover

The purpose of '2+1 = 3' is to assert that 2+1 equals 3.

Therefore "2+1" would denote an infinite number of things — Metaphysician Undercover

You got it exactly backwards. Our method does not lead to '2+1' denoting infinitely many things. '2+1' denotes exactly one thing. On the other hand, 2+1 is denoted infinitely many ways:

2+1 is denoted by '2+1'

2+1 is denoted by '3'

2+1 is denoted 'sqrt(9)'

2+1 is denoted by '((100-40)/3)-17'

etc.

and that would make interpretation impossible — Metaphysician Undercover

The method of interpretation in mathematics does quite fine, thank you (nothwithstanding wrinkles such as Lowenheim-Skolem).

equations would be absolutely useless because the right side would just be saying the exact same thing as the left side — Metaphysician Undercover

The equation is the statement that the left side stands for the exact same thing as the right side. That is very useful.

If we want to know how much a company did in sales, the accountant starts by seeing that the company got 500 dollars from Acme Corp., and 894 dollars from Babco Corp, and 202 dollars from Champco Corp. Then the accountant reports:

500+894+202 = 1596

It's useful to know that '500+894+202' names the same number as named by '1596'.

You'd never solve any problems that way, because the problem would be solved prior to making the equation. — Metaphysician Undercover

No, usually the problem is to find out or prove that the left side and right side are equal or not equal. Or to find another term to more easily represent, say, the left side.

From my sales reports I have 500+894+202. Then I follow a procedure to eliminate '+' and arrive at just one numeral: '1596;. And I conclude: 500+894+202 = 1596. So I then see that '500+894+202' and '1596' name the same number.

If you didn't know that the two sides signified the exact same thing already (meaning the problem is solved) you could not employ the equals sign. — Metaphysician Undercover

One wouldn't honestly claim to know that the equation is true until one worked it out that it is true. Or to find a right side without '+' in it, then first one might have to perform the addition on the left side. This doesn't vitiate anything I've said.

I did not say there is a circumstance in which '2' and '1' do not denote numbers.

'2+1' is a compound term made from the constants '2' and '1' and the operation symbol '+'.

'2' denotes a number. '1' denotes a number. '+' denotes an operation. '2+1' denotes the result of the operation + applied to the numbers 2 and 1. That result is a number. Therefore, '2+1' denotes a number.

The denotation of 'the father of Jane Fonda and Peter Fonda' is Henry Fonda. The denotation is not Jane Fonda nor Peter Fonda nor the sibling relation nor the process of fatherhood.

'the father of Jane Fonda and Peter Fonda' refers to one specific person, and that person is Henry Fonda.

The denotation of '2+1' is 3. The denotation is not 2 nor 1 nor the process of adding 1 to 2.

Another way of saying '2+1' is 'the sum of 2 and 1'.

And the sum of 2 and 1 is 3. It is not 2 nor 1 nor the process of adding 1 to 2. Rather the sum is the RESULT of the process of adding 1 to 2. The sum is not a process; it is a number.

Henry Fonda = the father of Jane Fonda and Peter Fonda. Henry Fonda IS the father of Jane Fonda and Peter Fonda.

'Henry Fonda' and 'the father of Jane Fonda and Peter Fonda' both refer to one identical person. Henry Fonda is identical with the father of Jane Fonda and Peter Fonda. There are not two different people - Henry Fonda vs. the father of Jane Fonda and Peter Fonda. There is one identical person with two ways of referring to him.

There are not two different numbers 2+1 vs. 3. There is one identical number with two ways of referring to it.

But your account of the meaning of mathematics is not compatible with the ordinary formulation of mathematics, so if your account were to have any consequence, then it would need to refer to some other formulation.
— GrandMinnow

As I said, if this point is of relevance then the discussion is pointless. — Metaphysician Undercover

You are repeating yourself without arguing specifically to the point I made. You argue by mere assertion. My point stands.

A contradiction is a statement and its negation. You have not shown any contradiction in what I said. The fact that '1', '2' and '2+1' each denote distinct numbers is not a contradiction.
— GrandMinnow

I can't believe that you do not understand the contradiction. — Metaphysician Undercover

Given your pattern of ignorance and confusion I can believe that you don't understand that you haven't shown a contradiction in my remarks. A contradiction implies both a statement and its negation. You have not shown how anything I've said implies both a statement and its negation.

Let' take the expression "2+1". Do the symbols "2" and "1" refer to distinct objects. If so, then there are two objects referred to by "2+1", and it is impossible, by way of contradiction, that "2+1" refers to only one object. — Metaphysician Undercover

You skipped the answer I gave to that already.

The sequence of steps is not the process, it is the description of the process. That this is an important distinction is evident from the fact that the very same process may be described in different ways, different steps, depending on how the process is broken down into steps. — Metaphysician Undercover

Since you have not given a mathematical definition of 'process', I am taking 'process in the sense of 'algorithm' or 'effective procedure'.

Mathematics addresses your point by recognizing that different processes may compute the same function.

Abstraction is simply how we interpret things — Metaphysician Undercover

There are two different senses, e.g. (1) "I think by means of abstraction" and (2) "My thinking has resulted in arriving at the abstract concept of blueness."

No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing.
— GrandMinnow

You agreed that they are different things which have the same result, or the same value. — Metaphysician Undercover

No, I definitely did not agree with that.

Indeed, I have explained for you that '4+2', '10-4' and '6' are not things that have results. They are NAMES, not numbers and not processes. Then, 4+2, 10-4, and 3 are the same number. What I do recognize is that, e.g., the process of adding 2 to 4 is different from the process of subtracting 4 from 10.

For about the fifth time:

4+2 is a number.

'4+2' is not a number; it is the name of a number.

If you simply refuse to understand the use/mention distinction, then you are doomed to continue in confusion.

Why do you want to say that adding 2 to 1 is the exact same thing as taking 3 from 6 — Metaphysician Undercover

Please stop ignoring the distinctions I have said multiple times already.

I said the RESULT of adding 2 to 1 is the same as the RESULT of subtracting 3 from 6. I do not say that the processes are the same.

why treat properties as if they are any sort of object? — Metaphysician Undercover

The abstraction called 'blueness' is an abstract object. 'Blueness' can be the subject of a sentence in the manner of a subject that refers to an object. For example, the previous sentence itself is one in which 'blueness' is the subject. And 'blueness' refers to the abstraction blueness.

I suspect that another big obstacle for you is that you don't understand that usually mathematics is extensional, not intensional.
— GrandMinnow

I've argued elsewhere that the axiom of extensionality is a falsity. — Metaphysician Undercover

The extensional nature of mathematics does not depend on the axiom of extensionality. They are different things. You just jump to the conclusion that because 'extension' is found in describing both things that one must depend on the other. You don't know what you're talking about.

In other words, equal things may be considered as the same thing. And that's clearly false. — Metaphysician Undercover

It's true by definition. You are welcome to define terminology in your own way, but I'm telling you in the meanwhile how the terminology is defined in mathematics.

https://plato.stanford.edu/entries/logic-intensional/

And a classic brief introduction to the subject is:

Introduction To Mathematical Logic, pages 1-9, by Alonzo Church

/

Most simply, ordinary mathematics is extensional, so substitutability of terms holds. That is, the principle of "substitute equals for equals" holds. That is, roughly put, for any terms T and S, and formula F, from T=S we may infer F[x|S] from F[x|T]. For example:


from

4 is even

we may infer

2+2 is even


But consider intensional contexts, such as this:


4 is even

and

4 = (((182/2)-1)/2)-66

and

Bob knows that 4 is even

therefore

Bob knows that (((182/2)-1)/2)-66 is even


The premises are true, but the conclusion is false if Bob doesn't know that 4 = (((182/2)-1)/2)-66.

Putting in 'knows' throws us into an intensional context where substitutability may fail.

/

I am not well versed beyond such basics as that, so for more on the subject I recommend the Stanford article and the passages in the Church book.

Aside from your lack of understanding of use/mention, I suspect that another big obstacle for you is that you don't understand that usually mathematics is extensional, not intensional.

'2+1' and '6-3' are different terms, so, even though extensionally they name the same number, the terms themselves have different intensional meanings. But ordinary mathematics works extensionally.

There have been proposals for formulating intensional mathematics. But I don't know whether your remarks would be pertinent to such formulations. Meanwhile, your remarks are completely off-base when it comes to mathematics as it is ordinarily studied.

When we say that there is an even number of chairs, this means that the group of chairs can be divided into two groups. — Metaphysician Undercover

When we say that n is even we mean that n is a natural number such there exists a natural number k such that n = 2k. But, yes, that does imply that a set with even cardinality has a partition of 2 sets with equal cardinality.

But the property, in everyday discussion (not even necessarily mathematics), of evenness of numbers is itself an abstraction. Properties are not things that are physical objects. Yes, physical objects have certain properties. But the properties themselves are abstractions and are not physical objects. You can't point to the property of blueness as a physical object. You can only point to certain blue things as having that property, but then you are pointing to those particular objects and not to the property itself.

we are invited to be critical of formulations — Metaphysician Undercover

But you don't know anything about the formulation of classical mathematics.

there is no need to offer an alternative formulation — Metaphysician Undercover

But your account of the meaning of mathematics is not compatible with the ordinary formulation of mathematics, so if your account were to have any consequence, then it would need to refer to some other formulation.

As I said, you equivocate: — Metaphysician Undercover

You misrepresented me when you wrote that I said that '1' and '2' do not refer to distinct objects. I said that they do refer to distinct objects. I'll say again, please do not misrepresent what I've said. I don't expect you to have the intellectual honesty to retract your claim about what I said, but I ask that at least you don't do it again.

And I have not equivocated.

Which is the case, do "1" and "2' each signify distinct numbers, or does "2+1" signify a number? You can't have it both ways because that's contradiction. — Metaphysician Undercover

A contradiction is a statement and its negation. You have not shown any contradiction in what I said. The fact that '1', '2' and '2+1' each denote distinct numbers is not a contradiction.

it is contradictory to say that "2+1" represents one number, because there are two numbers represented here. — Metaphysician Undercover

'2+1' has '1' and '2' as parts in the term. The term '1' denotes the number 1, the term '2' denotes the number 2, and the term '2+1' denotes the number 2+1, which is 3, which itself is also denoted by the term '3'. That is not a contradiction.

that the same quantitative value is predicated of the chairs in my dining room, and the musicians on that album, doesn't make that predicate into an object. — Metaphysician Undercover

I didn't say 6 is an object on account of 6 being the number of chairs or musicians. That was not my argument at all. You're confused about the structure of this discussion itself.

what is signified by "6-3" is not the same as what is signified by "2+1". You agree about this. — Metaphysician Undercover

No I do not. You are completely confused. (1) I would put aside 'signify' since you might have some special sense for it. I have mentioned 'refer to' or 'denote'. (2) And I do say that '6-3' and '2+1' denote or refer to the same number. I've said that about a dozen times already and you still don't recognize that that is what I say.

If you say that they have the exact same value, then we are using "equal" in the way I suggested. — Metaphysician Undercover

You are confused even about what you are saying yourself!

Saying that they have the exact same value is to say that they are identical.

Again '2+1' and '3' are not identical. They are different terms. But 2+1 and 6-3 are identical. They are the exact same number and they are the exact same referent of the terms '2+1', '6-3', and '3'.

you are very clearly talking about two distinct processes represented by "2+1", and "6-3". — Metaphysician Undercover

Again, 2+1 and 6-3 are not processes. They are numbers. They are the same number. They are the number 3. You skipped entirely my explanation that a process may be expressed as a certain kind of sequence of steps. The sequence of the steps is not the same as the last step itself, which is the result of the process. 2+1 and 6-3 denote the result. It is the same result.

Two distinct and different processes can have the same end result, and so those processes can be said to be equal. — Metaphysician Undercover

You have nearly everything backwards. A process is a sequence of steps. If one sequence of steps is different from another sequence of steps, then those are different processes even if their results are the same. Sequences are identical to each other if and only if every step in the sequences is the same. The sequence of steps in adding 1 to 2 is different from the sequence of steps in subtracting 3 from 6. But the results are the same for both. '1+2' and '6-3' do not denote processes; they denote numbers.

Does this imply, that in mathematics you judge a process according to the end result? — Metaphysician Undercover

No. Decidedly not. I just explained above, now for the second time.

If so, then how do you propose to judge an infinite process, which is incapable of producing an end result, like those referred to in the op? — Metaphysician Undercover

An infinite process that does not terminate is one that computes a recursively enumerable function. As to the notion of an infinite process that does terminate, I am not aware that this is a rigorous mathematical notion, though we are familiar with the philosophical notion of a supertask. I explained that also in a previous post to another poster.

You are invoking an imaginary object represented by "2", just like a theologian might invoke an imaginary object represented by "God". — Metaphysician Undercover

One difference is that mathematics works with algorithmically checkable axioms and rules of proof. Also, mathematics itself does not opine as to the ontological status of abstract objects but instead recognizes that we carry out mathematics with abstract reasoning regarding abstract objects - whatever we take such abstractions to be. Also, I tend to think that, strictly speaking, mathematicians could dispense even with the notion of objects, though it would make mathematical discussion clumsy.

Also, you have not answered how other abstractions could be acceptable, such as blueness or evenness or the state of happiness, etc.

You've been arguing that 4+2 is 6, and 10-4 is 6, and that there is potentially an infinite number of different things which are 6. — Metaphysician Undercover

No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing.

You just don't get the difference between names and the things named. That is a critical failure of understanding. You can't get past your confusions until you grasp the very simple distinction between a name and the thing that is named.

The number of chairs is referred to by "6". There is a specific quantity and that quantity is what is referred to with "6". I don't see where you get the idea of an object from here. There are six objects which form a group. The group is not itself the object being referred to, because the six are the objects. Therefore the quantity must be something other than an object or else we'd have seven, the six chairs plus the number as an object, which would make seven. — Metaphysician Undercover

The chairs are objects. And the mathematical object that is the number of chairs is the number 6. And the set of chairs also is an object, and it has cardinality 6.

You're just making an imaginary thing, like God, and handing a property, "even " to that thing — Metaphysician Undercover

Mathematical objects and mathematical properties are abstractions. They are not theological claims like the saying that there are gods. Also, properties like 'blueness' and 'evenness' are abstractions. You are free to reject that there are abstractions, but I use abstractions as basic in thought and reasoning.

When we use the symbol "2", we use it to refer to a group of two things. like chairs or something. — Metaphysician Undercover

We (not you though) understand the number 2 as not just the number of chairs or the number of any particular set of objects but also as a number onto itself.

Why assume that there is something other than a quantity, an object called 6? — Metaphysician Undercover

We prove from axioms that there is a unique object having a certain property, and we name it '6'. Why would we want to do that? Because it greatly facilities mathematics. I may refer to 6 itself rather than have to say "the number of chairs in Metaphysician Undercover's dining room".

In mathematics especially, names refer to things. The name '6' refers to something. It refers to a number. It refers to SSSSSS0. Mathematicians are undaunted by the fact that the thing named is an abstract object, not a concrete one.

where and how are we going to find this object?. — Metaphysician Undercover

We don't find it by a physical search. We find it by a mathematical activity in abstract reasoning.

you're claiming two nouns, 2 and 1, are one noun signified as "2+1". — Metaphysician Undercover

You are thoroughly mixed up, not just about mathematics, but about plain and simple things I've just posted.

I did not claim that 2 and 1 are nouns. I did not claim that 2 and 1 are signified by '2+1'.

2 is a number. 1 is a number.

'2' is a noun. '1' is a noun.

2+1 is a number.

'2+1' is a noun.

'2+1' does not denote 1.

'2+1' does not denote 2.

'2+1' denotes 2+1, and '2+1' denotes 6-3, and '2+1' denotes 3.

Learn use-mention, no matter what your philosophy is.

GM claims that in the context of "2+1" there is only one object referred, and "2" and "1" do not each refer to a distinct object. — Metaphysician Undercover

Please do not misrepresent what I said. I said explicitly that '1' and '2' do each refer to a distinct object. My remarks should not be victim to misrepresentation by you.

The issue I am looking at, is not how things are viewed by "ordinary mathematics", it is what is meant by the mathematical concepts. — Metaphysician Undercover

What is meant by whom? What is meant by mathematicians is not what is meant by you.

If we adhere to how things are viewed by mathematics, as if this is necessarily the correct view of things — Metaphysician Undercover

I haven't said here what is necessarily correct. There are formulations of mathematics - classical, constructivist, intuitionist, finitist - and then there are philosophical discussions about them. A formulation is not the same as the philosophical discussions about it. You are free to present a formulation (or at least an outline) of mathematics and then say philosophically what you mean by it. But lacking a formulation, I would take the context of a discussion of mathematics to be ordinary mathematics and not your unannounced alternative formulation.

No, "2" and "1" signify values. — Metaphysician Undercover

You're mixed up as to what I've said. Yes, I agree, and never disagreed, that '2' and '1' denote values.

Or do they sometime signify values and other times signify arguments — Metaphysician Undercover

This is another case in which your nearly total ignorance of mathematics results in your confusions.

Those numbers are both values themselves and also arguments applied to a function that in turn has a value for those arguments.

I see no way that a function, which is a process, could have a value. That's like saying that + has a value. — Metaphysician Undercover

You don't see a lot of things, because you refuse to even look at an introductory book on the subject.

Do you know what a function is? (In mathematics for grownups, there is a more rigorous definition of a function than 'process'.) Do you know how mathematics develops the subject of functions?

Meanwhile, I didn't say the function has a value. I said the function applied to arguments has a value.

It is the mathematical object that is the number of chairs, and is the number musicians on the album 'Buhaina's Delight', and is the value of the addition function for the arguments 4 and 2 ...
— GrandMinnow

I really don't know what you could possibly mean by this. — Metaphysician Undercover

It could not be more clear. 6 is the number of chairs in your dining room, and 6 is the number of musicians on the album 'Buhaina's Delight', and 6 is the number that is the value of the addition function for the arguments 4 and 2.

"2+1" means to put two together with one, and 2+1 equals "6-3", which means to take three away from six. — Metaphysician Undercover

We've gone over this multiple times already. 2+1 is the result of adding 2 and 1. 6-3 is the result of subtracting 3 from 6. The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6.

One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6.

2+1 is not a process. 2+1 is a number. It is the exact same number as 3, and the exact same number as 6-3. And '2+1' is a name of the number 2+1, and it is a name of the number 3, and it is a name of the number 6-3.

Meanwhile, a process in mathematics can be described as a certain kind of sequence of steps. Yes, the sequence of steps in adding 1 and 2 is different from the sequence of steps in subtracting 3 from 6. But the last entry in the sequence - the result - is the same. '2+1' and '6-3' are not names of processes; they are names of a number.

2+1 = 6-3. That is, 2+1 is 6-3.

'2+1' does not= '6-3'. But they are two different names of the same number.

No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)". — Metaphysician Undercover

I am telling you the terminology and framework of ordinary axiomatic mathematics. 'qualifier' is not the terminology used. Of course, you may set up your own terminology and framework, but the chance that it will make sense for ordinary axiomatic mathematics is slim since you don't know anything about ordinary axiomatic mathematics.

"+" represents an operation. So there are two distinct values, "2", and "1" represented, in "2+1", along with the operation represented by "+". — Metaphysician Undercover

(1) You are still making your use-mention mistake. Yes, '+' represents an operation and '2+1' is a representation of a value, but '2' and '1' are not values, they are representations of values.

Get it straight: The name of the object has quote marks and is not the same as the object. In cases where quote marks are used, we have a term that represents a value. When we mention the value itself, we don't use quote marks.

(2) As I explained, and as you ignored, + is the operation; 2 and 1 are the arguments; and 2+1 is the value of the function for those arguments. Yes, the operation is represented by '+; and the arguments are represented by '2' and '1'. And the value of the operation is represented by '2+1'. And that value is 2+1. Again, you need to learn basic use-mention.

that's a false assumption which I've discussed on many threads — Metaphysician Undercover

Your personal, confused, incoherent and uninformed views in that thread were demolished and shown to be confused, incoherent and uninformed.

You and I are equal, as human beings, but we are in no way identical with each other. "Equals" is clearly not another word for identical with. — Metaphysician Undercover

You are conflating the meaning of the world 'equal' in various other topics, such equality of rights in the law, with the more exact and specific meaning in mathematics. Human equality means that your rights are identical with my rights, and yes, that does not entail that you and I are identical. But in mathematics, which is ordinarily extensional, the equal sign '=' stands for identity. Again, you can use terminology in your own way for your own notions, but the chance that it will make sense for ordinary axiomatic mathematics is slim since you don't know anything about ordinary axiomatic mathematics.

Do you agree that a symbol has a meaning, which is not necessarily an object? — Metaphysician Undercover

Ordinary axiomatic mathematics is extensional. Each n-place operation symbol refers to a function on the domain of the interpretation, and functions are objects. The function might or might not be an object that is a member of the domain, but it is an object in the power set of the Cartesian product on the domain.

there is no need to assume that "2" or "3" represent objects. We'd have to look at how the symbols were being used, the context, to determine whether they represent objects or not. — Metaphysician Undercover

Not in ordinary mathematics where the numerals represent natural numbers. Or, in greater generality, any constant symbols, such as '1' and '2' are either primitive or defined, and in either case they represent members of the domain of interpretation.

As for defined symbols ('1' and '2' are more often defined rather than primitive), it is true that we cannot define the symbol without first proving the existence/uniqueness theorem. That is, we prove that there exists a unique object having a given property, then we define the symbol as standing for that unique object.

So 1 is the unique object that it is equal to the successor of 0. And 2 is the unique object that is equal to the successor of 1.

When I say that there are 6 chairs in my dining room, "6" refers to a number, but this is the number of chairs; the chairs are the objects and the number 6 is a predication. — Metaphysician Undercover

Ordinary mathematics does not view numbers as predictions.

The number is not an object — Metaphysician Undercover

It is the mathematical object that is the number of chairs, and is the number musicians on the album 'Buhaina's Delight', and is the value of the addition function for the arguments 4 and 2 ...

it is something I am saying about the chairs — Metaphysician Undercover

No what you are saying about the chairs is that the set of them has cardinality 6. You're not saying the chairs or the set of them has 6 or is 6. You are saying that the number of them is 6. 6 is not a property of chairs. Rather, a property of a (set of chairs) is that its cardinality is 6.

just like when I say "the sky is blue", blue is not an object. — Metaphysician Undercover

'blue' is an adjective, which is a certain kind of word, which is a linguistic object. It does not alone stand for an object. When we say the sky is blue, we say that the sky has the property of blueness. The sky is the object and blueness is the property. When we say that 2 is even, we mean that 2 has the property of being even. 2 is the object, and evenness is the property. When we say that 2+1 equals 3, we mean that that 2+1 and 3 as an ordered pair <2+1 3> are in the reflexive relation of equality.

'2' is a constant (a kind of "noun") in mathematics.

So you are comparing apples and oranges when you compare the noun '2' with the adjective 'blue'.

The correct analogy is:

With 'the sky is blue' we have the noun 'the sky' that stands for the sky, and 'blue' is an adjective, and 'is blue' stands for the property that holds for the sky.

With 'My dining room has six chairs', we have the noun 'my dining room' that stands for your dining room, and 'has six chair's' stands for the property that holds for your dining room.

With '2+1 = 3', we have the nouns '2+1' and '3', and '=' stands for the 2-place property of equality and indicates in the equation that the property of equality holds for the pair <2+1 3>.

/

I suppose it is an absolute given that you will never look at even page one of a book on the subject of mathematical foundations.

sqrt(2)" represents an operation — Metaphysician Undercover

sqrt is an operation. sqrt(2) is the object that is the result of the operation applied to the object 2. sqrt is the operation, and 2 is the argument to which the operation is applied.

+ is an operation. 2+1 is the object that is the result of the operation applied to the objects 2 and 1. + is the operation and 2 and 1 are the arguments to which the operation is applied.

what does it mean for a process to resolve to an object — Metaphysician Undercover

Operations are functions. A function is a certain kind of ordered pair. The result of an operation is simply the unique second coordinate for the ordered pair whose first coordinate is the argument.

we cannot produce the precise object which "sqrt(2)" is equivalent to — Metaphysician Undercover

We cannot finitely list the decimal expansion of sqrt(2). But sqrt(2) is a particular object. Also, what is important in this regard is not some object that the sqrt(2) is "equivalent to" (with whatever equivalence relation might be in mind) but rather with sqrt(2) itself.

quantitative value "2+1" is equivalent to a definite quantitative value represented by "3" — Metaphysician Undercover

Your use-mention is inconsistent there. Yes, '3' represents a value. But so also does '2+1'.

3 and 2+1 are values. They are the same value. Exactly the same. 3 = 2+1. 3 equals 2+1. 'equals' is another word for 'identical with'.

'3' and '2+1' are names that represent values. '3' and '2+1' are not equal. They are different names. But they name the same object. They are two different names for the same object.

Look up the subject of 'use-mention'.

Having a definite quantitative value is what makes the number an object — Metaphysician Undercover

A number is an object. If it's not an object, then what is it? If it is something that, according to you, might or might not be an object, then what is that something to begin with if not an object? How can we refer to something that is not an object?

the goal when using mathematics is to measure things — Metaphysician Undercover

Mathematics may be used for purposes other than measuring things.

which is to assign to them definite quantitative values — Metaphysician Undercover

We do mention a definite value when we mention sqrt(2). It doesn't have a finite decimal expansion, but it is a definite value.

algorithm (described completely with finite characters) which if executed to completion — Ryan O'Connor

Algorithms that execute to completion do so in a finite number of steps. As far as I know, what you may have in mind is not an algorithm but rather it is a supertask. I am not well informed about supertasks, so I don't know whether there is a rigorous mathematical definition of the notion or whether the notion is purely philosophical.

Cauchy Sequences — Ryan O'Connor

Equivalence classes of Cauchy sequences. This has been mentioned to you previously in this thread

why do we even need to assume that irrational 'numbers' exist? Why not assume that irrationals are the algorithms that we actually work with? — Ryan O'Connor

(1) We don't assume they exist. We prove they exist.

(2) I imagine that instead of constructing the real numbers, we could instead take the carrier set to be the set of algorithms. But limiting to only computable reals makes calculus a lot more complicated. Since calculus works with the irrationals included, and since it is more complicated for the calculus to regard the irrationals as algorithms and not as real numbers, we must weigh the advantages and disadvantages of both approaches. In particular, if we ask 'What is the length of the diagonal of a unit square?' we may answer 'sqrt(2)' rather than have to say 'Well, I can't tell you except there is an algorithm that computes successive approximations. Tell me what degree of accuracy would your like your approximation to be, and I'll tell you the answer to that degree of accuracy.' And I would say, 'Thanks, but I got a more succinct answer from the mathematician who said it is sqrt(2).'

And, by the way, if you show me an algorithm, then I may ask, 'What does it compute?' And how would you say what it computes without already having in mind that it computes approximations of ... wait for it ... sqrt(2). So, to get me sold on your algorithm, you would already have to presuppose that there is a thing that it approximates - and that thing is ... wait for it .. sqrt(2).

(3) There are proposed systems in constructivism, computationalism, and predicativism that may very well satisfy your desiderata. You only need to first inform yourself of the basics of the subject.

Most of this also has been mentioned to you previously in this thread.

I can accept that the square root operation is closed over the 'reals', but that doesn't mean it's closed over the real numbers. — Ryan O'Connor

'The reals' means 'the real numbers'.

We construct the set of real numbers. It doesn't make sense to debate whether a real number is a number. Mathematics doesn't have a universal definition of 'number'. Mathematics doesn't really involve questions of what is or is not a number. Instead there are many different number systems. Each number system has its carrier set. And we may ask whether certain objects or are or not in the carrier set of a given number system. The carrier set of the real number system is the set of real numbers (sometimes just called 'the reals). Every real number is a number in the sense that it is in the carrier set of the real numbers system.

A version of this information was provided to you earlier in this thread.

[Many philosophers of mathematics] simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry

My rough impression is that professionals in the field of philosophy of mathematics usually do know about mathematics. Which philosophers in, say, the last 85 years do you have in mind?

I didn't say it isn't perfectly fine English. I said you haven't properly identified the subject signified with "there", to which "exists an object" is predicated. — Metaphysician Undercover

Whatever you have in mind linguistically is irrelevant since the sentence is linguistically perfectly correct.

Even more simply:

There is a unique object whose square is 2.

or

A unique object exists such that its square is 2.

or

An object exists such that its square is 2 and no other object is such that its square is 2.

or

A unique object exists such that it has the property that its square is 2.

Etc.

All perfectly grammatical and sensible English.

I didn't write "E!x^". It doesn't make sense. I wrote "E!x x^2 = 2".

"There exists an object that has the property that its square is equal to 2" is perfectly fine English.

E!x x^2 = 2

is a theorem of ordinary mathematics.

Anyway, I made my point that existence is not a predicate..

"Existence" is a word which is being used here as a predicate. — Metaphysician Undercover

In casual discussion, mathematicians may say things like "the square root of 2 exists". But in a more careful mathematical context, we don't say that. Instead we say, "There exists a unique x such that x^2 = 2." Then we may apply the square root operator to refer to sqrt(2).

So, indeed, in careful mathematics 'existence' is not a predicate. In careful mathematics It is not even grammatical to use 'existence' as a predicate. Instead, there is an existential quantifier that is applied to a variable and a formula (usually with that variable free in the formula); the formula specifies a "property".

Symbolically, existence is not a predicate in which we would write "the square root of x has the property of existing:

E(sqrt(2))

That is not even grammatical.

Instead, we write:

Ex x^2 = 2

Which reads, "There exists an x such that x^2 = 2".

'Ex' is the quantifier, and 'x^2 = 2' is the formula specifying the property.

Then we also derive a uniqueness quantifier and write:

E!x x^2 = 2

Which reads, "There exists a unique x such that x^2 = 2".

And that justifies using

sqrt(2)

as a term.

I do have a problem with infinitesimals — Ryan O'Connor

Infinitesimals are made rigorous with non-standard analysis derived with techniques of model theory or with internal set theory.

the foundations — Ryan O'Connor

set theory — Ryan O'Connor

You don't know what set theory is.

You don't know about the symbolic logic in which set theory is formulated. You don't know what the language of set theory is. You don't know the axioms of set theory. You don't know the theorems of set theory and how they are derived from the axioms. You don't know the definitions in set theory. You don't know how set theory develops numbers and mathematics. You don't know how set theory axiomatizes calculus and other mathematics of the sciences. You don't know the purpose, motivation, and role of the axiomatic method. You don't know about constructive, intuitionist, predicativist, or finitist altermatives to classical mathematics.

You don't know anything about it.

Yet you have persistent critiques of it.

How do you do it?

If set theory properly lies at the foundation of mathematics then (I believe) it should have no loose threads (e.g. paradoxes). — Ryan O'Connor

I addressed that already. You blew right past it.

Do you believe that infinite processes cannot be completed? — Ryan O'Connor

I don't think in a framework of "infinite processes being completed or not completed". The notion of "an infinite process being completed or not completed" is not a notion I find meaningful; I am not burdened with it.

Moreover I already addressed this with regard to set theory:

Set theory itself (at least at the level of this discussion), as formal mathematics, does not say "an infinite process can be completed". Set theory doesn't even have vocabulary that mentions "completion of infinite processes". And the assumptions of set theory are the axioms. There is no axiom of set theory "an infinite process can be completed". — GrandMinnow

But you blow right past that. (You're too busy with things like explaining that I am free to ignore you, and extending my little metaphor of a 'trail' into an overblown conceit that is becoming ludicrous).

infinite sum — Ryan O'Connor

An infinite sum is the limit of a function. It is the unique number such that the terms of the sequence converge to that number.

The actual mathematics does not say "process" or "process completion". It doesn't need to be saddled with it. If you find problems with the intuitive view of an infinite sum as the completion of a process, then it's your intuitive framework that is problematic, not mine, since I don't have to resort to that framework and, as far as I know, mathematics may be understood without it.

So much more to unpack:

Imagine having a discussion with a child — Ryan O'Connor

Imagine a discussion among intelligent and educated adults.

If they ask a question, one way of addressing it is to add layers of complexity to the issue such that it is beyond their grasp — Ryan O'Connor

Among these educated and intelligent adults, if they post their opinions on certain matters that bear upon technical considerations, then one may point out where those technical points have been misconstrued and offer relatively concise corrections and explanations. If this is beyond the grasp of any of these educated and intelligent adults then they can consult any of a number of books that explain at a level any intelligent person can understand by simply beginning with chapter one and reading forward.

pile on a dozen textbooks — Ryan O'Connor

You have been occasionally exaggerating my points in order to knock them down. And you use other variations of the strawman argument, most saliently to me in the very first remarks in your video on Zeno's paradox, as I pointed out. (Though you have since retracted in this thread, I don't know whether you intend to correct the video itself).

To the point, I did not "pile on a dozen textbooks". You asked me about education. I gave you a list of three (plus some optional supplements) introductory undergraduate textbooks for a three step sequence: Symbolic Logic, Set Theory, Mathematical Logic. And I don't say that one has to have such a background merely to ask questions, speculate, ruminate, or convey ideas on the subject. Rather, my point is that when your philosophizing moves into matters that bear on technical points in mathematics, and you mangle those points or talk past right past them, then it's appropriate to point that out. And I recommended a few books in response to your question about education.

and say 'ask me when you know what you're talking about' — Ryan O'Connor

I never said that or anything equivalent to it.

I'm here to learn — Ryan O'Connor

Perhaps you are, but that's not all. You've also here for other people to be impressed with what you say.

We don't need gatekeepers — Ryan O'Connor

I'm not a gatekeeper in the sense of saying that people may not post whatever they want to post. You can post as you like; I don't try to stop you. Meanwhile, I hope you are not a gatekeeper saying what I may post, including criticisms of your posts. And my purposes in posting are not determined by what you think a forum needs or doesn't need.

we need people to help the litterers learn how not to litter — Ryan O'Connor

I have suggested ways you can abate your littering.

feel free to ignore my messages — Ryan O'Connor

I already responded to your sophomoric protest that I may ignore you. Of course I feel free to ignore you, and I also feel free not to ignore you. Your personal preference in the matter is not relevant to me. When you post, others may reply or not reply arbitrarily at their own prerogative.

if you're inclined to help then I welcome it — Ryan O'Connor

I've offered you help already. I've given you explanations at a pretty simple and straightforward level. But you blow right past most of the key points in those explanations. And I've given you a list of three books that constitute a truly splendid introduction to the subject on which you are posting. I am not even a mathematician, but at least I have made myself familiar with a number of books on the subject. If you are sincere about learning and benefiting from what I know, then the very best you could do is to accept my expertise in book collecting and get hold of the books I mentioned. Instead you take umbrage at the offer and whine as if you've been unduly sandbagged.

I don't think you will enjoy us talking informally about potential [problems with the current philosophical foundations for math] — Ryan O'Connor

If you don't want to talk informally, or if you want to disregard Zeno's paradox due to its informal presentation that is fine — Ryan O'Connor

I am, as time permits, interested in philosophy of mathematics and informal discussion about mathematics and the philosophy of mathematics. And Zeno's paradox is of course important in the philosophy and history of mathematics. But this is the point you keep missing: When the informal discussion bears upon, or especially critiques, the actual mathematics and the actual foundational formalizations, then it is critical not to speak incorrectly, especially from ignorance, about the actual mathematics, formal theories, and the developments in set theory and mathematical logic. I surmise that you, like cranks, find poring through the actual technical development to be onerous but you prefer to opine about it in ignorance anyway. This is witnessed by the fact that no matter how many times one suggests to a crank that he consult the actual writings on the subject, he will never even look at chapter one.

It's worth noting that the challenges in the first post of this thread have been met. But hell if I know whether the poster understands that by now.

1. Is flush with critiques of a subject while he is unwilling to inform himself of the basics of that subject by even reading an introductory textbook on it. No matter how many times it is pointed out that he is terribly confused on basic points, he will never just pick up a book on the subject.

2. Keeps confusing technical points. But he keeps eliding the corrections presented by resorting to the cop-out "I'm only talking about it philosophically", even though the philosophizing is a critique of a technical subject. Or he just skips over the decisive corrections, as instead he replies by adding even more diverting tangents.

3. Projects onto others that they are not giving him a fair chance, that they don't try to understand him. Yet he keeps skipping over the actual explanations from others as to where he is confused, incorrect, and ill-informed.

4, Thinks he is presenting an innovative alternative in the subject. Yet he ignores the work already done in the subject over the recorded history of man - work by people who have dedicated truly incredible intellectual curiousness, creativity, rigor, and industriousness, while responsibly submitting their work to the most exacting standards of the peer-review method. This includes even ignoring serious work in alternatives to classical foundations - work that may be aligned with what he himself is ineptly stumbling to convey. The literature blooms with finitist, computationalist, constructivist and myriad other alternatives. Yet he won't inform himself about them.

5. Finally resorts to umbrage and the sophomoric instruction to ignore his posts, presumably then not to comment on them. Even though it was just explained that at least one motive in commenting on his posts is to not leave his falsehoods, misconceptions, and confusions uncorrected. Also, anyone should understand that it is the prerogative of posters arbitrarily to read and comment on whatever they want and that saying "then don't read my posts" is likely a doomed instruction anyway.

6. His misconceptions center on the usual crank bugbear: infinity.

So 1 through 6. But do we dare say 'crank'?

Do you want to live in a country where the 'scenic trails' are exclusive to the 'privileged rich'? — Ryan O'Connor

I appreciate that threads are open to posting by both well informed and less informed posters. That doesn't entail that misinformation, misconception, and confusion should not be called out for what it is.

You are trying to find a way to reject my ideas without understanding them — Ryan O'Connor

I have explained exactly how certain of your ideas are ill-conceived and how you disservice the subject on which on which you opine while ignorant of its basics.

You asked me to look at your videos. Upon looking at one, I found that near the very start, you made a claim that "mathematicians begin with an assumption that an infinite process can be completed." I asked you to please say what specific statement by a mathematician you have in mind so that we can understand its context and to see how it fits your claim as to what mathematicians assume. Your critique of classical mathematics itself makes assumptions about classical mathematics - most of them quite ill-founded. You are the one who is critiquing ideas about mathematics without understanding them.

Again, set theory rises to the challenge of providing a formal system by which there is an algorithm for determining whether a sequence of formulas is indeed a proof in the system. So whatever you think its flaws are, that would have to be in context of comparison with the flaws of another system that itself rises to that challenge. — GrandMinnow

That is a central point that I have made twice now. You have not responded.

A possibility occurs to me: When people who don't study the actual mathematics of set theory hear about such things as the axiom of infinity or encounter the notion of an infinite set, they don't grasp it except on their own terms. They can only grasp it by imposing their own explanation of what they think it means, and with that imposed explanation they declare that the notion, such as that of an infinite set, is wrong. Then they never move past that stubborn misunderstanding, no matter how many times one suggests starting out by actually reading the basics of the subject.

In particular, the notion of a 'process' is imposed on axioms that don't mention 'process' at all. Granted, some mathematicians do mention stages of set building and levels attained (or, for example, see Potter's book in which he actually makes an equivalent set theory axiomatization as a theory of levels), for the purpose of providing an intuitive or philosophical framework work for thinking about the mathematics, while the actual mathematics is not itself liable for whatever difficulties may be found in such intuitive or philosophical frameworks.

If there is no way to reinterpret the Axiom of Infinity — Ryan O'Connor

As I alluded previously, your "reinterpret an axiom" has no apparent meaning (surely not rigorous) other than as a vague personal notion. Axioms are formal syntactic objects; you have not provided any meaningful sense of what in general a "reinterpretation" of an axiom is.

Moreover, do you even know what the axiom of infinity is? Do you even know the basics of the first order language of set theory in which the axiom of infinity is formulated?

When working with ZF, we are always dealing with finite statements. — Ryan O'Connor

Formulas of ZF are finite sequences of symbols. But ZF itself is a certain infinite set of formulas. And an axiomatization of ZF is a certain infinite set of formulas. Yes, any given formula of ZF is finite, and any given proof in ZF is finite (indeed, not just finite, but algorithmically checkable). But the study of ZF goes on to considerations of infinite sets of formulas (and while the set of axioms itself is infinite, it is algorithmically checkable whether any given formula is or is not an axiom). .

isn't a forum like this a good place to discuss [half-baked ideas]? — Ryan O'Connor

The personal context you present is based in gross ignorance of the subject. I don't opine as to all discussions, but this discussion engenders an unfortunate trail of waste product - gross misinformation, misunderstanding, and confusion. Sometimes people who have an appreciation of the subject wish not to see such otherwise beautiful scenic trails left without being de-littered.

If you were sincere about this subject, rather than blindly swinging about flimsy claptrap, you would familiarize yourself with the basics of the subject. I know nothing about marine molecular biology, so I don't go into a thread saying "the notion of cell structures in mollusks needs to be reinterpreted." I don't go into a thread about neuroepistemology and say "the very notion of a neural network is not acceptable to me; I propose instead an inside-out interpretation instead of the classical outside-in framework."

Do limits require the existence of infinite sets? — Ryan O'Connor

The study of limits in ordinary calculus involves, among other things, functions on real intervals, which are infinite. Infinite sets, intervals, domains, ranges, functions, et. al are all over calculus.

Do you believe that there are any paradoxes related to the set theoretic axiomatization of mathematics — Ryan O'Connor

Paradoxes are related to set theory and foundations, of course. But formal set theory does not have paradoxes; it has or does not have actual formal contradictions.

how much education should a person have before initiating a discussion on a philosophy forum like this? — Ryan O'Connor

I don't have a general answer to such a question. But I would suggest for you this undergraduate sequence:

Symbolic Logic (suggest: 'Logic: Techniques Of Formal Reasoning - Kalish, Montague & Mar; supplemented by chapter 8 ('Theory Of Definition') in 'Introduction To Logic' - Suppes)

Set Theory (suggest: 'Elements Of Set Theory' - Enderton, electively supplemented by 'Axiomatic Set Theory' - Suppes)

Mathematical Logic (suggest: 'A Mathematical Introduction To Logic - Enderton)

What resolution of Zeno's paradox are you satisfied with? Limits can be used to describe a process of approaching a destination but they cannot describe arriving there. So how does one arrive at some new destination? — Ryan O'Connor

Zeno's paradox (at least as it is usually presented) is not a formal mathematical problem, but instead is a challenge to certain intuitive explanations of certain observable phenomena. 'arriving' and 'destination' are not, in this context, mathematical terms. Set theory is not responsible for disentangling every everyday common notion. Meanwhile, in mathematics we speak of the limit of a function at a point. It is quite clear and, as far as I know, it works for solving certain scientific problems.

Are your and fishfry's posts in agreement? — Ryan O'Connor

You may reasonably ascribe to me only what I post myself. I imagine that holds for any poster.

videos — Ryan O'Connor

I watched the one about Zeno's paradox up to the point you said, "mathematicians begin with an assumption [that] [an] infinite process can be completed". What exact particular quote by a mathematician are you referring to? Please cite a quote and its context so that one may evaluate your representation of it in context, let alone your generalization about what "mathematicians [in general] begin with as an assumption",

Set theory itself (at least at the level of this discussion), as formal mathematics, does not say "an infinite process can be completed". Set theory doesn't even have vocabulary that mentions "completion of infinite processes". And the assumptions of set theory are the axioms. There is no axiom of set theory "an infinite process can be completed".

You're talking about how you'd like mathematics to be, but you do it entirely in a castles-in-the-air manner without regard to even a minimal understanding of the mathematical context. Philosophizing about mathematics is fine. But when the philosophizing concerns actual mathematical concepts, then, unless there is an understanding of the actual mathematics and the demands of deductive mathematics, that philosophizing is bound to end as heap of half-baked gibberish.

Ordinary calculus does use infinite sets. As I mentioned, one can provide finitistic or other alternative axiomatizations, but, as I said, to fairly evaluate the advantages and disadvantages of such alternatives, we would have to know really what those axiomatizations are. You should understand this point well: The set theoretic axiomatization of mathematics is very straightforward, easy to understand, and eventually yields precise formulations for the notions of the mathematics of the sciences. Meanwhile, as best I can find, many finitistic alternatives are either much more complicated, harder to grasp, and possibly fail rigor.

If you are sincerely interested in the subject, even from a philosophical point of view, you should learn the set theoretic foundations and then also you could learn about alternative foundations that bloom in the mathematical landscape.

I'm not up to untangling all of your comments, but here are some points:

it's unreasonable to expect a formal theory to be perfected in isolation. — Ryan O'Connor

I didn't say that initially in a discussion the formal aspect has to be perfect. It just has to be reasonably coherent and credible.

When Descartes developed analytic geometry, I suspect that he didn't present axioms — Ryan O'Connor

I haven't read the mathematical papers of Descartes, but I suspect that he presented some basic principles and reasoned deductively from them. Then, over centuries, the deductive principles and methods of mathematics became more and more sharpened, as we eventually articulated the notion of formality as recusiveness and algorithmic effectiveness.

Zeno's paradox, derivative paradox, dartboard paradox) are not paradoxical with this view because this view is void of actual infinity — Ryan O'Connor

And neither is there a Zeno's paradox with set theoretic infinity.

with a few lines of code I can create a program to list all natural numbers even though it is impossible for the program to be executed to completion — Ryan O'Connor

We already have that concept. It's called 'recursive enumerability'.

I believe that if we abandon Platonism by replacing the actual infinities with potential infinities that the mathematics will stand. This is what I mean when I say it's a philosophy issue, not a mathematical issue. — Ryan O'Connor

I wouldn't begrudge philosophical objections to the notion of infinity. My point though is that one does not have to be platonist to work with theorems that are "read off" in natural language as "there exists an infinite set". The axiom that is (nick)named 'the axiom of infinity' does not mention 'infinity' and, for formal purposes, use of the adjective 'is infinite' can just as well be dispensed in favor of a purely formally defined 1-place predicate symbol.

With the 'Whole-from-Parts' view we can't put brackets around everything and call it math. There is no set of all sets. We cannot talk about division by zero. We can't avoid Gödel statements. — Ryan O'Connor

Division by zero can be handled by the Fregean method of definition. And I addressed Godel previously; you don't know what you're talking about with regard to Godel. Morevover, as you mention what you consider to be flaws in classical mathematics, as I said before, you have not offered a specific alternative that we could examine for its own flaws. Again, set theory rises to the challenge of providing a formal system by which there is an algorithm for determining whether a sequence of formulas is indeed a proof in the system. So whatever you think its flaws are, that would have to be in context of comparison with the flaws of another system that itself rises to that challenge.

Mr. Public: One of the drawbacks of the current vaccines is that you have to stick a needle in the arms of people and many people don't like that.

Scientist: True. So what's your alternative?:

Mr. Public: I think we can do it with pills instead.

Scientist: What's your formulation? Where are your trials?

Mr. Public: I don't have those. I'm just approaching it from philosophy.

there can also be an unmeasured 'potential' state where a proposition is neither true nor false — Ryan O'Connor

The context of modern logic and mathematics involves formal axiomatization. Anyone can come up with all kinds of philosophical perspectives on mathematics and even come up with all kinds of ersatz informal mathematics itself. But that does not in itself satisfy the challenge of providing a formal system in which there is objective algorithmic verification of the correctness of proof (given whatever formal definition of 'proof' is in play). I mentioned this before, but you ignored my point. This conversation is doomed to go nowhere in circles if you don't recognize that what you're talking about is purely philosophical and does not (at least yet) provide a formalization that would earn respect of actual mathematicians in the field of foundations.

There is actual work in formal constructivist, finitist, and computationalist, or multi-valued mathematics, but you don't reference or commit to any specific such formulation. So we don't have a basis for evaluating the merits of your notions compared to those of classical mathematics on the level playing field of "I'll show you my system, explicitly, without vernacular vagueness and you show me yours."

Isn't 'not a number' a reasonable option to be included in the outcome space of the proof √2. — Ryan O'Connor

Again, please show your system of axioms and rules by which one may make an evaluation of such circumstances.

Why do we need all mathematical objects to actually exist? — Ryan O'Connor

If it's an object, then it exists. At the very least, in formal terms, there is an existence theorem.

Interpretation 1: Any finite list of primes is incomplete.
Interpretation 2: There exist infinitely many primes.

These two interpretations may seem equivalent but they're not because the second makes an unjustified leap to assert the existence of an infinite set. — Ryan O'Connor

The justification is from axioms from which we prove that there exists an infinite set. You are free to reject, or be uninterested in, those axioms, but in the meanwhile you haven't said what your axioms are. And you are even free to reject the axiomatic method itself, or be uninterested in it. But then this conversation would be at a hard impasse between, on the one hand, mathematicians who understand the benefit of formal axiomatics and require that mathematical proposals be backed up not just by homespun philosophizing, and, on the other hand, you.

I believe ZF and Peano arithmetic just need to be reinterpreted. — Ryan O'Connor

ZF and PA are formal systems. They are interpreted by the method of models. Whatever you have in mind by a reinterpretation of a formal system is not stated by you. However for you to state such a thing would require that you do know the basics of mathematical logic that is the context of such systems.

I simply think we shouldn't interpret ZF in certain platonic ways. — Ryan O'Connor

Fine. We don't need to. However, the mathematics itself stands whether the mathematician regards it platonistically or not.

Real numbers are never used in applied mathematics. Every number that we have ever used in a calculation is rational. — Ryan O'Connor

That is patently false. Of course, usually physical measurements, as with a ruler, involve manipulation of physical objects themselves and as humans we can't witness infinite accuracy in such circumstances. But that doesn't meant that mathematical calculations are limited in that way.

ZF just needs to be reinterpreted. This is largely an issue of philosophy, not mathematics. — Ryan O'Connor

ZF is a formal system. In ZF we prove that there exists a system, which we denote as 'the real number system' and we prove that it is a complete ordered field.

Aren't you beginning your proof with an assumption, that irrationals are numbers? — Ryan O'Connor

Yes, as I stated explicitly, "Supposing there is a real number x such that x^2 = 2."

I gave that as a supposition in the context of separating the two questions I mentioned, But, meanwhile "there is a real number x such that x^2 =2" is proven from the ordinary axioms.

Why can't it simply be an algorithm? — Ryan O'Connor

If you show us your actual system for mathematics, then we could evaluate its heuristic advantages or disadvantages compared with classical mathematics. But just saying "it's an algorithm not a number" is a an informal thesis, not an argument.

My view is that we don't need to 'decide' every mathematical statement because 'undecided' is a valid state. — Ryan O'Connor

Fine. Then your remark to the other poster about undecidability is not pertinent. By the way, in classical mathematics a statement that is undecided is not undecided simpliciter but rather it is undecided relative to a given system.

My view is in total agreement with the foundations of calculus. — Ryan O'Connor

No it's not. Clearly.

Understanding the sense of mathematical existence statements - such as the existence of irrational numbers - does require subscribing to mathematical platonism. — GrandMinnow

I meant to write "[...] does not require [...]." I edited my post just now upon reading your post and realizing that I mistakenly left out the word 'not'.

He did not prove that √2 is an irrational number. — Ryan O'Connor

The proof that there is a real number x such that x^2 = 2 comes later in the history of mathematics. It is found in many a textbook in introductory real analysis. This is desirable, for example, so that we can easily refer to a particular real number that is the length of the diagonal of the unit square.

won't there always be undecidable statements? — Ryan O'Connor

Do you think your version of non-axiomatic mathematics "decides" every mathematical statement? There are undecidable statements in the mathematics of the plain counting numbers themselves. Undecidability is endemic to the most basic mathematics even aside from the real numbers as a complete ordered field. So how does your version of non-axiomatic mathematics "decide" all mathematical questions? By polling among the billions of known existing people (and examining just completed computer computations) whether they have such and such mathematical answers or computations in their minds or in their computer output at some given time?

Why is it necessary to have a number system which is complete? — Ryan O'Connor

This was answered by another poster. I would add that a complete ordered field is the ordinary foundation for calculus, which is used for the ordinary mathematics for the physical sciences.

And we can do exact arithmetic using any rational number — Ryan O'Connor

Arithmetic, sure. But you haven't shown how to do exact calculus without irrational numbers.

The mainstream approach to giving the number √2 existence requires us to assert the existence the Platonic Realm (which I equate with an infinite computer), — Ryan O'Connor

Mathematical platonism (roughly put) is the view that mathematical objects exist independent of consciousness of them and that mathematical propositions are true or false independent of conscious determination. Understanding the sense of mathematical existence statements - such as the existence of irrational numbers - does not require subscribing to mathematical platonism.

We separate two questions:

(1) Is there a real number x such that x^2 = 2?

(2) Supposing there is a real number x such that x^2 = 2, is that real number rational or irrational? (Note that 'irrational' simply means, by definition, 'not rational').

Proof supplied to answer (1) depends on certain axioms. Usually, these are the set theoretic axioms used to prove the existence of the real numbers as a complete ordered field. The real numbers as a complete ordered field provide a foundation for the mathematics of the physical sciences. Axiomatization is a desirable approach to mathematics, as it provides an explicit objective algorithmic standard by which anyone may judge of a purported mathematical proof whether it is indeed a correct proof, as opposed to subjective standards such as what happens to be or not be in the mind at any given time of some human being or another. So, if you reject certain set theoretic axioms, then we may ask: What are your alternative axioms?

Perhaps (I don't opine for the purposes of this particular post) proof supplied to answer (2) may be called into question constructively by rejecting the dichotomy that a real number is either rational or it is not rational. However, as to proof by contradiction, the proof that sqrt(2) is irrational is of this form [let P be any proposition]: We wish to prove it is not the case that P (in particular, we wish to prove that it is not the case that the sqrt(2) is rational). We suppose it is the case that P. We then derive a contradiction. We conclude that it is not the case that P. That is a constructive proof form.

That is not to be confused with the non-constructive proof form: We wish to prove it is the case that P. We suppose it is not the case that P. We then derive a contradiction. We conclude that it is the case that P.

As to the overall contradiction form, the irrationality of sqrt(2) is of the former, constructive, form.

There are myriad ways to spout nonsense, fallacy, and misinformation such as yours, but fewer distinctive ways to state accuracies. So I am at a disadvantage relative to you, as your errant posts may have greater variety while mine eventually become repetitious. But I will respond yet again, while knowing that there is a good chance that eventually I'll give up trying to convince you that you are not prepared to discuss mathematical logic without having at least read an introductory textbook.

You are confusing the symbol sets of the systems that the incompleteness theorem are about with the symbol set of any system in which the incompleteness theorem itself is proven. The former is what is in question here. (However both are countable anyway.)

The Godel-Rosser incompleteness theorem is that there is not a formal, consistent, "arithmetically adequate" system that yields a negation-complete theory. Systems with uncountable symbol sets are not generally considered to be formal. I explained why in a previous post. Indeed, moreover, the proof of the incompleteness theorem relies on assigning a unique natural number to each symbol, thus the symbol set must be countable.

You do a disservice by posting, in many threads, widely incorrect nonsense while you are not familiar with the basics of the subject that are found in introductory textbooks. GET A TEXTBOOK IN MATHEMATICAL LOGIC.