Well, the square root operation is closed over real numbers i.e. a square root of a real number has to be a real number — TheMadFool
You've got to be kidding. Think about this statement for, say, five seconds. :roll: — jgill
What's the square root of the real number -4? — jgill
Why did God go with the complex numbers and not the real numbers?
Years ago, at Berkeley, I was hanging out with some math grad students -- I fell in with the wrong crowd -- and I asked them that exact question. The mathematicians just snickered. "Give us a break -- the complex numbers are algebraically closed!" To them it wasn't a mystery at all.
But to me it is sort of strange. I mean, complex numbers were seen for centuries as fictitious entities that human beings made up, in order that every quadratic equation should have a root. (That's why we talk about their "imaginary" parts.) So why should Nature, at its most fundamental level, run on something that we invented for our convenience?
Answer: Well, if you want every unitary operation to have a square root, then you have to go to the complex numbers...
Scott: Dammit, you're getting ahead of me!
Alright, yeah: suppose we require that, for every linear transformation U that we can apply to a state, there must be another transformation V such that V^2 = U. This is basically a continuity assumption: we're saying that, if it makes sense to apply an operation for one second, then it ought to make sense to apply that same operation for only half a second. — PHYS771 Lecture 9: Quantum - Scott Aaronson
This is why I think the number 3 can exist but not the 'number' sqrt(2). We never actually work with irrational 'numbers', we only work with their algorithms or rational number approximations. So 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
sqrt(2)" represents an operation — Metaphysician Undercover
what does it mean for a process to resolve to an object — Metaphysician Undercover
we cannot produce the precise object which "sqrt(2)" is equivalent to — Metaphysician Undercover
quantitative value "2+1" is equivalent to a definite quantitative value represented by "3" — Metaphysician Undercover
Having a definite quantitative value is what makes the number an object — Metaphysician Undercover
the goal when using mathematics is to measure things — Metaphysician Undercover
which is to assign to them definite quantitative values — 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. — GrandMinnow
Your us-mention is inconsistent there. Yes, '3' represents a value. But so also does '2+1'. — GrandMinnow
'equals' is another word for 'identical with'. — GrandMinnow
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? — GrandMinnow
No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)". — Metaphysician Undercover
"+" represents an operation. So there are two distinct values, "2", and "1" represented, in "2+1", along with the operation represented by "+". — Metaphysician Undercover
that's a false assumption which I've discussed on many threads — Metaphysician Undercover
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
Do you agree that a symbol has a meaning, which is not necessarily an object? — Metaphysician Undercover
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
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
The number is not an object — Metaphysician Undercover
it is something I am saying about the chairs — Metaphysician Undercover
just like when I say "the sky is blue", blue is not an object. — Metaphysician Undercover
Have we really proved the existence of irrational numbers? That's the name of this thread.All proofs of the existence irrational numbers (that I'm aware of) are proofs by contradiction. For example, we assume that √2 can only be 1) a rational number or 2) an irrational number. Since we've proved that √2 is not a rational number we conclude that it's an irrational number. Is it possible that this is a false dichotomy? — Ryan O'Connor
(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. — GrandMinnow
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. — GrandMinnow
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. — GrandMinnow
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. — GrandMinnow
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
You are saying that the number of them is 6. — GrandMinnow
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. — GrandMinnow
With '2+1 = 3', we have the nouns '2+1' and '3', and '=' stands for the 2-place predicate of equality, and indicates in the equation that the predicate of equality holds for the pair <2+1 3>. — GrandMinnow
No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)". — Metaphysician Undercover
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
If we adhere to how things are viewed by mathematics, as if this is necessarily the correct view of things — Metaphysician Undercover
No, "2" and "1" signify values. — Metaphysician Undercover
Or do they sometime signify values and other times signify arguments — Metaphysician Undercover
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
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
"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
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
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
You're just making an imaginary thing, like God, and handing a property, "even " to that thing — Metaphysician Undercover
When we use the symbol "2", we use it to refer to a group of two things. like chairs or something. — Metaphysician Undercover
Why assume that there is something other than a quantity, an object called 6? — Metaphysician Undercover
where and how are we going to find this object?. — Metaphysician Undercover
you're claiming two nouns, 2 and 1, are one noun signified as "2+1". — Metaphysician Undercover
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. — GrandMinnow
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. — GrandMinnow
I said explicitly that '1' and '2' do each refer to a distinct object. — GrandMinnow
2+1 is a number. — GrandMinnow
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. — GrandMinnow
The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6. — GrandMinnow
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. — GrandMinnow
Mathematical objects and mathematical properties are abstractions. They are not theological claims like the saying that there exists a God. Also, properties like 'blueness' and 'evenness' are abstractions. You are free to reject that there are abstractions, but I use abstractions as basic in human reasoning. — GrandMinnow
We prove from axioms that there is a unique object having a certain property, and we name it '6'. — GrandMinnow
we are invited to be critical of formulations — Metaphysician Undercover
there is no need to offer an alternative formulation — Metaphysician Undercover
As I said, you equivocate: — Metaphysician Undercover
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
it is contradictory to say that "2+1" represents one number, because there are two numbers represented here. — Metaphysician Undercover
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
what is signified by "6-3" is not the same as what is signified by "2+1". You agree about this. — Metaphysician Undercover
If you say that they have the exact same value, then we are using "equal" in the way I suggested. — Metaphysician Undercover
you are very clearly talking about two distinct processes represented by "2+1", and "6-3". — Metaphysician Undercover
Two distinct and different processes can have the same end result, and so those processes can be said to be equal. — Metaphysician Undercover
Does this imply, that in mathematics you judge a process according to the end result? — Metaphysician Undercover
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
You are invoking an imaginary object represented by "2", just like a theologian might invoke an imaginary object represented by "God". — Metaphysician Undercover
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
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
'2+1' and '6-3' are different terms, so, even though extensionally they name the same number, — GrandMinnow
the terms themselves have different intensional meanings. — GrandMinnow
But you don't know anything about the formulation of classical mathematics.
...
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
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
A process is a sequence of steps. — GrandMinnow
Also, you have not answered how other abstractions could be acceptable, such as blueness or evenness or the state of happiness, etc. — GrandMinnow
No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing. — GrandMinnow
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. — GrandMinnow
Properties are not things that are physical objects. — GrandMinnow
I suspect that another big obstacle for you is that you don't understand that usually mathematics is extensional, not intensional. — GrandMinnow
That is, the principle of "substitute equals for equals" holds. — GrandMinnow
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
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
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
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
Abstraction is simply how we interpret things — Metaphysician Undercover
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
Why do you want to say that adding 2 to 1 is the exact same thing as taking 3 from 6 — Metaphysician Undercover
why treat properties as if they are any sort of object? — Metaphysician Undercover
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
In other words, equal things may be considered as the same thing. And that's clearly false. — Metaphysician Undercover
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. — GrandMinnow
The denotation of '2+1' is 3. The denotation is not 2 nor 1 nor the process of adding 1 to 2. — GrandMinnow
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. — GrandMinnow
Where is the result of the operation denoted? — Metaphysician Undercover
already denoted — Metaphysician Undercover
Otherwise there would be absolutely no purpose to the "=" because "2+1" on its own does not say 3. — Metaphysician Undercover
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
Therefore "2+1" would denote an infinite number of things — Metaphysician Undercover
and that would make interpretation impossible — Metaphysician Undercover
equations would be absolutely useless because the right side would just be saying the exact same thing as the left side — Metaphysician Undercover
You'd never solve any problems that way, because the problem would be solved prior to making the equation. — Metaphysician Undercover
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
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