Mathematical-Philosophical Question

[jaybird0827]

In Algebra 2: An Incremental Development (Saxon, John H., Second Edition, 1991) I just came across this. He is defining "imaginary number".

"Thus, on the number line, we can locate both +sqrt(4) and -sqrt(4) as shown,

(graphic illustration shows the two numbers, +2 and -2 graphed on the number line and also labeled by their respective equivalents +sqrt(4) and -sqrt(4)).

but we cannot locate +sqrt(-4) or -sqrt(-4) on the number line. For this reason, unfortunately, we call sqrt(-4) an imaginary number. We say unfortunately, because all numbers are ideas and thus all numbers are really imaginary."

Emphasis original. How might we critique the statement "All numbers are ideas and thus all numbers are imaginary." from a Biblical perspective?

 

[FenderPriest]

"All numbers are ideas and thus all numbers are imaginary" is false because God is both one and many, This establishes the eternal, and basic nature of numbers in the ontology of the Trinity. Numbers exist because God is three in one - God can fundamentally count within himself, and thus the impression of numerical values are a part of the imageo deo and revelation.

Frege has a good book on the subject of numbers, Foundations of Arithmetic. While he's not a Christian, his destruction of the idea that numbers are categories of the physical world, and Mills proposition of the foundation of arithmetic is classic.

 

[FenderPriest]

"All numbers are ideas and thus all numbers are imaginary" is false because God is both one and many, This establishes the eternal, and basic nature of numbers in the ontology of the Trinity. Numbers exist because God is three in one - God can fundamentally count within himself, and thus the impression of numerical values are a part of the imageo deo and revelation.

Frege has a good book on the subject of numbers, Foundations of Arithmetic. While he's not a Christian, his destruction of the idea that numbers are categories of the physical world, and Mill's proposition of the foundation of arithmetic is classic.

Sorry to repost on this, but I was thinking about the statement, and was more concerned about the frame of the statement than the values: "All ... ideas ... are imaginary." This could be stated like this:

"All numbers are ideas and thus all numbers are imaginary"

Where P = numbers
Q = Ideas
I = Imaginary

All P are Q
(All Q are I)
Thus: All P are I

The second proposition is left unstated, and undefended (going off of what you’re presented). This seems to be the fundamental error in the writer’s thinking: all ideas are imaginary. This is nonsensical. Are the ideas imaginary in being made up? Are they imaginary in their existence? Doesn’t imaginary thought require an existent being? Doesn’t this contradict the proposition? The fundamental question that is merely assumed, and thus left undefended, is the nature of thought and whether it exists, and what relation it has to the outside world. Thus, the question over the existence of numbers is an epistemological question in nature. Anyhow, these were my further thoughts on the problem proposed.

~Jacob

 

[Civbert]

...
Emphasis original. How might we critique the statement "All numbers are ideas and thus all numbers are imaginary." from a Biblical perspective?

Not without knowing exactly what he means by "imaginary" - or more important, what he didn't mean. I wouldn't assume he meant that numbers are not real. I think he meant that imaginary" numbers are as real as "rational" numbers. He is simply pointing out that it is unfortunate that we have decided to call this type of numbers "imaginary" as if they were any less real.

 

[VictorBravo]

...
Emphasis original. How might we critique the statement "All numbers are ideas and thus all numbers are imaginary." from a Biblical perspective?

Not without knowing exactly what he means by "imaginary" - or more important, what he didn't mean. I wouldn't assume he meant that numbers are not real. I think he meant that imaginary" numbers are as real as "rational" numbers. He is simply pointing out that it is unfortunate that we have decided to call this type of numbers "imaginary" as if they were any less real.

Right. I think he is getting at the problem with terminology. Under at least one definition of "imaginary", of course all ideas are imaginary because they can exist in a mind as a form of image (a loose image, perhaps, but something is there in the mind representing the idea). Yet "imaginary" also can mean "not real".

So when I think "square root of two" I can "see" in my mind either the square root symbol over the symbol for two, or I can picture a diagonal line of a square with unit 1 sides, etc. It's imaginary in that sense because there is no real square or line in front of me.

But for the square root of a negative number, I can only picture the symbol, not the diagonal of a square. It's still imaginary, but I can't see it as real. So it is unfortunate to call it an imaginary number.

 

[jaybird0827]

...
Emphasis original. How might we critique the statement "All numbers are ideas and thus all numbers are imaginary." from a Biblical perspective?

Not without knowing exactly what he means by "imaginary" - or more important, what he didn't mean. I wouldn't assume he meant that numbers are not real. I think he meant that imaginary" numbers are as real as "rational" numbers. He is simply pointing out that it is unfortunate that we have decided to call this type of numbers "imaginary" as if they were any less real.

I agree with the idea that an "imaginary number" is no less real than a "real number" and that the author is clear on this issue. No problem.

I'm trying to deal with his assumption that "all ideas are imaginary". The idea seems foreign to Scripture - can we verify that? If so, how is it opposed to Scripture, and how do we answer it.

 

[jaybird0827]

...
Emphasis original. How might we critique the statement "All numbers are ideas and thus all numbers are imaginary." from a Biblical perspective?

Not without knowing exactly what he means by "imaginary" - or more important, what he didn't mean. I wouldn't assume he meant that numbers are not real. I think he meant that imaginary" numbers are as real as "rational" numbers. He is simply pointing out that it is unfortunate that we have decided to call this type of numbers "imaginary" as if they were any less real.

Right. I think he is getting at the problem with terminology. Under at least one definition of "imaginary", of course all ideas are imaginary because they can exist in a mind as a form of image (a loose image, perhaps, but something is there in the mind representing the idea). Yet "imaginary" also can mean "not real".

So when I think "square root of two" I can "see" in my mind either the square root symbol over the symbol for two, or I can picture a diagonal line of a square with unit 1 sides, etc. It's imaginary in that sense because there is no real square or line in front of me.

But for the square root of a negative number, I can only picture the symbol, not the diagonal of a square. It's still imaginary, but I can't see it as real. So it is unfortunate to call it an imaginary number.

What you said in paragraph 1 is particularly helpful - the double meaning of the word "imaginary".

 

[jaybird0827]

"All numbers are ideas and thus all numbers are imaginary" is false because God is both one and many, This establishes the eternal, and basic nature of numbers in the ontology of the Trinity. Numbers exist because God is three in one - God can fundamentally count within himself, and thus the impression of numerical values are a part of the imageo deo and revelation.

Frege has a good book on the subject of numbers, Foundations of Arithmetic. While he's not a Christian, his destruction of the idea that numbers are categories of the physical world, and Mill's proposition of the foundation of arithmetic is classic.

Sorry to repost on this, but I was thinking about the statement, and was more concerned about the frame of the statement than the values: "All ... ideas ... are imaginary." This could be stated like this:

"All numbers are ideas and thus all numbers are imaginary"

Where P = numbers
Q = Ideas
I = Imaginary

All P are Q
(All Q are I)
Thus: All P are I

The second proposition is left unstated, and undefended (going off of what you’re presented). This seems to be the fundamental error in the writer’s thinking: all ideas are imaginary. This is nonsensical. Are the ideas imaginary in being made up? Are they imaginary in their existence? Doesn’t imaginary thought require an existent being? Doesn’t this contradict the proposition? The fundamental question that is merely assumed, and thus left undefended, is the nature of thought and whether it exists, and what relation it has to the outside world. Thus, the question over the existence of numbers is an epistemological question in nature. Anyhow, these were my further thoughts on the problem proposed.

~Jacob

Jacob - that last paragraph is more along the line of what I'm after. There is much more at stake here than mathematical jargon.

 

[VictorBravo]

Not without knowing exactly what he means by "imaginary" - or more important, what he didn't mean. I wouldn't assume he meant that numbers are not real. I think he meant that imaginary" numbers are as real as "rational" numbers. He is simply pointing out that it is unfortunate that we have decided to call this type of numbers "imaginary" as if they were any less real.

Right. I think he is getting at the problem with terminology. Under at least one definition of "imaginary", of course all ideas are imaginary because they can exist in a mind as a form of image (a loose image, perhaps, but something is there in the mind representing the idea). Yet "imaginary" also can mean "not real".

So when I think "square root of two" I can "see" in my mind either the square root symbol over the symbol for two, or I can picture a diagonal line of a square with unit 1 sides, etc. It's imaginary in that sense because there is no real square or line in front of me.

But for the square root of a negative number, I can only picture the symbol, not the diagonal of a square. It's still imaginary, but I can't see it as real. So it is unfortunate to call it an imaginary number.

What you said in paragraph 1 is particularly helpful - the double meaning of the word "imaginary".

I'm glad it helped. I'll exand just a little more. I agree that in common usage, "imaginary" means unreal or fanciful. But the definition of imagination also usually includes something like this:

"The formation of a mental image of something that is neither perceived as real nor present to the senses.
The mental image so formed.
The ability or tendency to form such images."

So the statement "all ideas are imaginary" isn't necessarily a problem as long as we are not saying they are fanciful, but rather we are saying that we are thinking of something (real) that is not present to the senses. That, I think fits fine with our Biblical understanding of concepts and truth--we don't taste, see, or smell them, but we know they are real.

 

[Zenas]

COGITO ERGO SUM!

Ideas necessarily lend themselves to existence because physicality is required for the formulation of ideas.

*shrug* Maybe that doesn't make sense though.

 

[jaybird0827]

I'm glad it helped. I'll exand just a little more. I agree that in common usage, "imaginary" means unreal or fanciful. But the definition of imagination also usually includes something like this:

"The formation of a mental image of something that is neither perceived as real nor present to the senses.
The mental image so formed.
The ability or tendency to form such images."

So the statement "all ideas are imaginary" isn't necessarily a problem as long as we are not saying they are fanciful, but rather we are saying that we are thinking of something (real) that is not present to the senses. That, I think fits fine with our Biblical understanding of concepts and truth--we don't taste, see, or smell them, but we know they are real.

Wow! I was thinking of an example like this earlier: I pass through the kitchen and I see that there are 3 apples in the fruit bowl. I remember that as I leave the room. The thought is real, and the 3 apples in the bowl are also a reality.

 

[jaybird0827]

COGITO ERGO SUM!

Ideas necessarily lend themselves to existence because physicality is required for the formulation of ideas.

*shrug* Maybe that doesn't make sense though.

It sort of does? Can you think of an example? What about "abstract" ideas?

 

[VictorBravo]

COGITO ERGO SUM!

Ideas necessarily lend themselves to existence because physicality is required for the formulation of ideas.

*shrug* Maybe that doesn't make sense though.

Heh, I don't think Descartes was arguing that he thought, therefore he was physical. He was not an empiricist.

Plus, that would run into trouble with Isaiah 55:8 among others: "For my thoughts are not your thoughts, neither are your ways my ways, saith the LORD."

 

[Zenas]

Ah that is true.

 

[Brian Bosse]

Hello Everyone,

My post is divided into three sections: (1) All Ideas are the Product of a Mind, (2) Imaginary Numbers, and (3) a Caveat.

All Ideas are the Product of a Mind

How might we critique the statement "All numbers are ideas and thus all numbers are imaginary." from a Biblical perspective?

Jacob is right to point out that there is a latent syllogism contained in the above statement…

Premise 1: All numbers are ideas.
Premise 2: All ideas are imaginary. (Assumed but unstated.)
Conclusion: All numbers are imaginary.

Jacob says that the author’s statement is unbiblical and that premise 2 is “nonsensical”. I think Jacob is wrong on both points. First off, we need to try and read the author in the best light. As such, we ought to try and understand his statement in a consistent manner. If one understands the term ‘imaginary’ in the sense of “product of a mind,” then I see no problem with the argument. All ideas are products of a mind. Secondly, the Biblical worldview would tell us that numbers exist in the mind of God. That is to say, the ground for the existence of the number ‘1’ is that this abstract entity is part of the thinking of God. Understood within this context, then I can affirm the author’s statement.

Imaginary Numbers

An imaginary number is defined to be a “number” whose square is a negative real number. For instance, i*i is defined to be -1. Even though imaginary numbers are defined with terms and concepts we can understand, we cannot conceive of such entities - much like we cannot conceive of a square circle. If this is the case for us, it might be the same for God. (It could be that in our finitude we do not have the ability of conceive of such an entity, but God might be able to.) If God cannot conceive of such an entity, then on this basis one could argue that such an entity cannot exist.

A Caveat

A philosopher might argue that some of the most beautiful equations in mathematics utilize imaginary numbers (for instance, e^(i*pi)+1=0). They might also point out the fruitful use of imaginary numbers in fields such as quantum mechanics. That is to say, they might point out that there are mathematical models utilizing imaginary numbers that seemingly correlate to the real world. From a pragmatic perspective, they would argue this lends support to the existence of such numbers.

My personal position is that imaginary numbers, although utilized in very surprising ways, are inconceivable to us and to God. As such, they do not exist. I hold to this belief lightly. I do not want to presume too much on what God can and cannot know. If such numbers do exist, it is only because God can rationally conceive of such numbers.

Sincerely,

Brian

 

[Civbert]

Hello Everyone,

...
Imaginary Numbers

An imaginary number is defined to be a “number” whose square is a negative real number. For instance, i*i is defined to be -1. Even though imaginary numbers are defined with terms and concepts we can understand, we cannot conceive of such entities - much like we cannot conceive of a square circle. If this is the case for us, it might be the same for God. (It could be that in our finitude we do not have the ability of conceive of such an entity, but God might be able to.) If God cannot conceive of such an entity, then on this basis one could argue that such an entity cannot exist.

A Caveat

A philosopher might argue that some of the most beautiful equations in mathematics utilize imaginary numbers (for instance, e^(i*pi)+1=0). They might also point out the fruitful use of imaginary numbers in fields such as quantum mechanics. That is to say, they might point out that there are mathematical models utilizing imaginary numbers that seemingly correlate to the real world. From a pragmatic perspective, they would argue this lends support to the existence of such numbers.

My personal position is that imaginary numbers, although utilized in very surprising ways, are inconceivable to us and to God. As such, they do not exist. I hold to this belief lightly. I do not want to presume too much on what God can and cannot know. If such numbers do exist, it is only because God can rationally conceive of such numbers.

Sincerely,

Brian

Good post. One encounter I've had with imaginary numbers was in physics when we had to use complex notation to express relationships in electrical impedance. I can't recall all the details, but we had to work with results that were complex (part real and part imaginary). I think this also came up when dealing with AC circuits and electric phases. And I think this all had to do with the relationship between magnetic and electric fields which act counter to each other.

The point I want to make is that although "imaginary numbers" seem inconceivable, this is because if we try to conceive of them in a physical 3-D world, they don't work. We can "visualize" obtaining a square root of a number - we can show this geometrically. But this approach fails when we need to deal with the square roots of negative numbers. In fact, negative numbers are entirely abstract. You can not diagram a negative space because any attempt would have positive dimensions. There will always be a contradiction between the physical representation and the mathematical definition. All physical space is positive.

However, the mathematical relationships involving negative numbers or negative roots are entirely logical. I don't think this is a square-circle issue. There is nothing necessarily contradictory about them. They simply highlight how abstract mathematical concepts can be. However, abstract does not equate with unreal. In the case of imaginary numbers, we can show a real-world relationship in electromagnetic physics. Even if we can not perfectly visualize the relationships.

A very interesting subject. Philosophy is everywhere, even in the hard sciences.

 

[Civbert]

The wikipedia article on "imaginary numbers" is worth reading. Here's a quote:

Many other mathematicians were slow to believe in imaginary numbers at first, including Descartes who wrote about them in his La Géométrie, where the term was meant to be derogatory.