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How Do Numbers Exist?

tarkovsky
Posts: 212
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1/14/2013 4:19:44 PM
Posted: 3 years ago
I think anyone with more than just a skin deep interest in either philosophy or mathematics might find this an interesting and important question. How do numbers exist? I feel quite comfortable saying that numbers don't have a corporeal existence; I've never encountered in my sensuous world, the number 1 or the constant e. Surely I've observed the physical characteristics of nature, some of which I might be able to describe with a number or some mathematical relation. However, perhaps a more technical and as yet a more demanding question would be how do mathematical objects exist?

My intuition tells me that mathematical objects exist conceptually. Numbers don't exist corporeally and there is no higher mathematical plane of reality from which this physical world derives. In fact, a number is a just a more exact telling of the more general notion of quantity. That said, numbers do not precisely represent the corporeal facts of which the natural world is comprised; all mathematics and numbers can do are stand as an approximation, an invention, a human interpretation of the brute facts of the nature.

For fear of starting off this discussion with what may already be an overly long post I'll stop here.
phantom
Posts: 6,774
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1/14/2013 4:28:37 PM
Posted: 3 years ago
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?
"Music is a zen-like ecstatic state where you become the new man of the future, the Nietzschean merger of Apollo and Dionysus." Ray Manzarek (The Doors)
000ike
Posts: 11,196
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1/14/2013 4:32:18 PM
Posted: 3 years ago
I feel like "exist" is one of those halfway ambiguous, halfway meaningless words. Whenever we claim that anything exists, we need to establish what we mean by it first.

Exist can mean that the object occupies physical space, or it can mean for something to occur, or for something to simply be a part of reality. And claiming that certain things exist can make abundant sense or no sense depending on which meaning is being implied. Consider the question, "do the laws of physics exist?" The answer is yes, if we're referring to existence in the sense of anything concerned with reality - as the "laws of physics" is just a name that describes operations of reality. The same goes for numbers. If existence was used in the sense of the prior definitions, then neither numbers nor laws can exist.
"A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel Foucault
ConservativePolitico
Posts: 8,210
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1/14/2013 4:33:50 PM
Posted: 3 years ago
We put values to a set number of actual objects. They're descriptions. How do colors exist? Same thing. We created number systems to describe a set of values that exist in the real world.

If you asked me how many apples I had, how could I respond without numbers? If I didn't have numbers I'd probably make up a descriptor to let you know exactly how many I have which would then eventually lead to a numerical system.
Apeiron
Posts: 2,446
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1/14/2013 4:39:58 PM
Posted: 3 years ago
At 1/14/2013 4:19:44 PM, tarkovsky wrote:
I think anyone with more than just a skin deep interest in either philosophy or mathematics might find this an interesting and important question. How do numbers exist? I feel quite comfortable saying that numbers don't have a corporeal existence; I've never encountered in my sensuous world, the number 1 or the constant e. Surely I've observed the physical characteristics of nature, some of which I might be able to describe with a number or some mathematical relation. However, perhaps a more technical and as yet a more demanding question would be how do mathematical objects exist?

My intuition tells me that mathematical objects exist conceptually. Numbers don't exist corporeally and there is no higher mathematical plane of reality from which this physical world derives. In fact, a number is a just a more exact telling of the more general notion of quantity. That said, numbers do not precisely represent the corporeal facts of which the natural world is comprised; all mathematics and numbers can do are stand as an approximation, an invention, a human interpretation of the brute facts of the nature.

For fear of starting off this discussion with what may already be an overly long post I'll stop here.

I think nominalism and Platonism is out. But the view that numbers are just mere useful fictions for describing reality, I think, is more tenable given the Peaneo axioms they rest upon.

For instance, we can make any thriller fiction we like if we set up in the story certain axioms that are accepted among the audience and so forth. But so say there's a real world wherein Jason murders everyone, is to start doing some pretty scary metaphysics ;-)
Apeiron
Posts: 2,446
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1/14/2013 4:40:48 PM
Posted: 3 years ago
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?
Apeiron
Posts: 2,446
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1/14/2013 4:41:33 PM
Posted: 3 years ago
At 1/14/2013 4:39:58 PM, Apeiron wrote:
At 1/14/2013 4:19:44 PM, tarkovsky wrote:
I think anyone with more than just a skin deep interest in either philosophy or mathematics might find this an interesting and important question. How do numbers exist? I feel quite comfortable saying that numbers don't have a corporeal existence; I've never encountered in my sensuous world, the number 1 or the constant e. Surely I've observed the physical characteristics of nature, some of which I might be able to describe with a number or some mathematical relation. However, perhaps a more technical and as yet a more demanding question would be how do mathematical objects exist?

My intuition tells me that mathematical objects exist conceptually. Numbers don't exist corporeally and there is no higher mathematical plane of reality from which this physical world derives. In fact, a number is a just a more exact telling of the more general notion of quantity. That said, numbers do not precisely represent the corporeal facts of which the natural world is comprised; all mathematics and numbers can do are stand as an approximation, an invention, a human interpretation of the brute facts of the nature.

For fear of starting off this discussion with what may already be an overly long post I'll stop here.

I think nominalism and Platonism is out. But the view that numbers are just mere useful fictions for describing reality, I think, is more tenable given the Peaneo axioms they rest upon.

For instance, we can make any thriller fiction we like if we set up in the story certain axioms that are accepted among the audience and so forth. But so say there's a real world wherein Jason murders everyone, is to start doing some pretty scary metaphysics ;-)

Also, I think the tell-tail sign if things exist is whether or not they're causally potent.
tarkovsky
Posts: 212
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1/14/2013 4:42:24 PM
Posted: 3 years ago
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

The are precise in representing themselves, but they cannot precisely represent something else. I think this sort of speaks to the ascribed belief of Feynman where he explains in his famous lecture that one day mathematics will not be necessary in creating a complete description of nature. That the laws of physics will be articulated without mathematics.

If that interests you, you can check Feynman's lecture entitled "The Relation of Mathematics and Physics" for free. Particularly, the fifth partition of the video will be of interest with respect to this discussion. Normally I eschew relying so heavily on sources but I think, in this case, I'd be doing the larger community a favor posting this link here. In any case, I believe the lecture can be watched purely on it's own merits.
http://research.microsoft.com...|d71e62e2-0b19-4d82-978b-9c0ea0cbc45f||
phantom
Posts: 6,774
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1/14/2013 4:45:49 PM
Posted: 3 years ago
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.
"Music is a zen-like ecstatic state where you become the new man of the future, the Nietzschean merger of Apollo and Dionysus." Ray Manzarek (The Doors)
tarkovsky
Posts: 212
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1/14/2013 4:47:00 PM
Posted: 3 years ago
At 1/14/2013 4:32:18 PM, 000ike wrote:
I feel like "exist" is one of those halfway ambiguous, halfway meaningless words. Whenever we claim that anything exists, we need to establish what we mean by it first.

Exist can mean that the object occupies physical space, or it can mean for something to occur, or for something to simply be a part of reality. And claiming that certain things exist can make abundant sense or no sense depending on which meaning is being implied. Consider the question, "do the laws of physics exist?" The answer is yes, if we're referring to existence in the sense of anything concerned with reality - as the "laws of physics" is just a name that describes operations of reality. The same goes for numbers. If existence was used in the sense of the prior definitions, then neither numbers nor laws can exist.

Well I wasn't asking if numbers existed but how they exist. I clarified I believed they had a conceptual existence which of course was left without rigorous definition. I assumed most people would have interpreted it as I meant it which was existing as nothing more than a thought, or a way of thinking.
phantom
Posts: 6,774
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1/14/2013 4:47:03 PM
Posted: 3 years ago
At 1/14/2013 4:42:24 PM, tarkovsky wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

The are precise in representing themselves, but they cannot precisely represent something else. I think this sort of speaks to the ascribed belief of Feynman where he explains in his famous lecture that one day mathematics will not be necessary in creating a complete description of nature. That the laws of physics will be articulated without mathematics.

If that interests you, you can check Feynman's lecture entitled "The Relation of Mathematics and Physics" for free. Particularly, the fifth partition of the video will be of interest with respect to this discussion. Normally I eschew relying so heavily on sources but I think, in this case, I'd be doing the larger community a favor posting this link here. In any case, I believe the lecture can be watched purely on it's own merits.
http://research.microsoft.com...|d71e62e2-0b19-4d82-978b-9c0ea0cbc45f||

Okay, thanks.
"Music is a zen-like ecstatic state where you become the new man of the future, the Nietzschean merger of Apollo and Dionysus." Ray Manzarek (The Doors)
000ike
Posts: 11,196
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1/14/2013 4:50:36 PM
Posted: 3 years ago
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

*Not intended to pile on from Apeiron's comment. This is a completely different criticism

Where did this dichotomy come from - numbers and what they represent? Numbers are names for certain states of things in reality...the same way "apple" is not different from the red/green/yellow juicy thing that we eat.
"A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel Foucault
Apeiron
Posts: 2,446
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1/14/2013 4:50:41 PM
Posted: 3 years ago
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

So two apples are necessary?

Doesn't 2+2=4 true in all possible worlds?
tarkovsky
Posts: 212
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1/14/2013 4:59:09 PM
Posted: 3 years ago
At 1/14/2013 4:50:36 PM, 000ike wrote:
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

*Not intended to pile on from Apeiron's comment. This is a completely different criticism

Where did this dichotomy come from - numbers and what they represent? Numbers are names for certain states of things in reality...the same way "apple" is not different from the red/green/yellow juicy thing that we eat.

Added to that, what does root two represent? Moreover, some mathematical objects are 'discovered' or 'created' (whatever everyone is most comfortable with saying) with respect to the purely theoretical. Negative roots were mathematically relevant long before we could determine any worldly analog. Same goes for negative numbers.
tarkovsky
Posts: 212
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1/14/2013 6:19:23 PM
Posted: 3 years ago
At 1/14/2013 4:33:50 PM, ConservativePolitico wrote:
We put values to a set number of actual objects. They're descriptions. How do colors exist? Same thing. We created number systems to describe a set of values that exist in the real world.

If you asked me how many apples I had, how could I respond without numbers? If I didn't have numbers I'd probably make up a descriptor to let you know exactly how many I have which would then eventually lead to a numerical system.

I see where you've come from with this, and I think 000ike might coming from a similar place. Though I agree that numbers probably emerged as a description of some natural state of the world, I can't say that I therefore believe numbers are just descriptions of natural states.

The mathematical Platonist brings up an interesting point, arguing that many mathematical objects emerge more as a discovery, where our object is purely an abstraction and we haven't the faintest clue to what extent it can describe some physical state of affairs. I think the popular example is the Mandelbrot set. Bear with me as I try to explain.

There are certain mathematical objects, such as the Mandelbrot Set, which gives pause to even the most ardent opponent of mathematical Platonism. Taken at face value, any knowing person would find it hard to bring themselves to say that the Mandelbrot Set was an invention. In fact, the overwhelming complexity in calculation is probably the only reason why it emerged only some twenty or thirty years ago. Though the set is merely a simple sequence of numbers, this simple sequence gives rise to infinite complexity which makes one ask the question; to what extent is this object mathematician-dependent? In fact the set itself is beyond comprehension insofar as it is infinitely complex, but the general form of the sequence is quite easily comprehended (z -> z^2+c). It feels as though it was always just there, independent of some mathematician, that is to say, the Mandelbrot Set and its properties exist whether there is a mathematician there to say it does or not. That complex numbers, though seemingly merely an invention who's properties extended to nothing more than root of negative one, somehow gives rise independently to all these other properties such as boundedness in the Argand Plane. Such properties were not included when we initially invented them, nor were they intended. They simply arose without us even knowing it and we're only coming to find out about these properties now.

That said, I'm still of the opinion that none of this means that there is some mathematical reality that we catch glimpses of and know of only in part. In fact, I think this only means that there are, no doubt, profound relationships in our conceptual world. Alas, that infinity, like number and quantity, is a human invention whereas the mathematician might consider it a sort of refuge for his phantasms.
phantom
Posts: 6,774
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1/14/2013 8:49:01 PM
Posted: 3 years ago
At 1/14/2013 4:50:41 PM, Apeiron wrote:
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

So two apples are necessary?

No. Not what I meant.

Doesn't 2+2=4 true in all possible worlds?

Yes, but numbers don't have to exist in all possible worlds. What the equation represents, holds true to reality necessarily but the concepts of numbers don't except in all worlds because concepts are something agents invent/conceive.
"Music is a zen-like ecstatic state where you become the new man of the future, the Nietzschean merger of Apollo and Dionysus." Ray Manzarek (The Doors)
Apeiron
Posts: 2,446
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1/14/2013 9:26:22 PM
Posted: 3 years ago
At 1/14/2013 8:49:01 PM, phantom wrote:
At 1/14/2013 4:50:41 PM, Apeiron wrote:
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

So two apples are necessary?

No. Not what I meant.

Doesn't 2+2=4 true in all possible worlds?

Yes, but numbers don't have to exist in all possible worlds. What the equation represents, holds true to reality necessarily but the concepts of numbers don't except in all worlds because concepts are something agents invent/conceive.

Well that's contradictory, right? If numbers hold in all possible worlds, then they're necessary. But if they're mind-dependant, then they can't be in a contingent mind, unless of course they're contingent. Which isn't what you said, thus by your own reason universals must be grounded in a necessary mind. Wouldn't you agree?
phantom
Posts: 6,774
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1/14/2013 9:31:24 PM
Posted: 3 years ago
At 1/14/2013 9:26:22 PM, Apeiron wrote:
At 1/14/2013 8:49:01 PM, phantom wrote:
At 1/14/2013 4:50:41 PM, Apeiron wrote:
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

So two apples are necessary?

No. Not what I meant.

Doesn't 2+2=4 true in all possible worlds?

Yes, but numbers don't have to exist in all possible worlds. What the equation represents, holds true to reality necessarily but the concepts of numbers don't except in all worlds because concepts are something agents invent/conceive.

Well that's contradictory, right? If numbers hold in all possible worlds, then they're necessary. But if they're mind-dependant, then they can't be in a contingent mind, unless of course they're contingent. Which isn't what you said, thus by your own reason universals must be grounded in a necessary mind. Wouldn't you agree?

No. What equations express are necessarily true but the numbers in the equation do not exist necessarily because the concept of "number" is just a useful thing we invented. The truth of the equation exists independent of a mind but the concepts used to express the equation are mind-conceived and the concepts are not necessary.
"Music is a zen-like ecstatic state where you become the new man of the future, the Nietzschean merger of Apollo and Dionysus." Ray Manzarek (The Doors)
Apeiron
Posts: 2,446
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1/14/2013 9:36:08 PM
Posted: 3 years ago
At 1/14/2013 9:31:24 PM, phantom wrote:
At 1/14/2013 9:26:22 PM, Apeiron wrote:
At 1/14/2013 8:49:01 PM, phantom wrote:
At 1/14/2013 4:50:41 PM, Apeiron wrote:
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

So two apples are necessary?

No. Not what I meant.

Doesn't 2+2=4 true in all possible worlds?

Yes, but numbers don't have to exist in all possible worlds. What the equation represents, holds true to reality necessarily but the concepts of numbers don't except in all worlds because concepts are something agents invent/conceive.

Well that's contradictory, right? If numbers hold in all possible worlds, then they're necessary. But if they're mind-dependant, then they can't be in a contingent mind, unless of course they're contingent. Which isn't what you said, thus by your own reason universals must be grounded in a necessary mind. Wouldn't you agree?

No. What equations express are necessarily true but the numbers in the equation do not exist necessarily because the concept of "number" is just a useful thing we invented. The truth of the equation exists independent of a mind but the concepts used to express the equation are mind-conceived and the concepts are not necessary.

I'm talking about universal truths now. Those axioms that are necessarily true, which such arithmetic are predicated upon.
phantom
Posts: 6,774
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1/14/2013 9:39:53 PM
Posted: 3 years ago
At 1/14/2013 9:36:08 PM, Apeiron wrote:
At 1/14/2013 9:31:24 PM, phantom wrote:
At 1/14/2013 9:26:22 PM, Apeiron wrote:
At 1/14/2013 8:49:01 PM, phantom wrote:
At 1/14/2013 4:50:41 PM, Apeiron wrote:
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

So two apples are necessary?

No. Not what I meant.

Doesn't 2+2=4 true in all possible worlds?

Yes, but numbers don't have to exist in all possible worlds. What the equation represents, holds true to reality necessarily but the concepts of numbers don't except in all worlds because concepts are something agents invent/conceive.

Well that's contradictory, right? If numbers hold in all possible worlds, then they're necessary. But if they're mind-dependant, then they can't be in a contingent mind, unless of course they're contingent. Which isn't what you said, thus by your own reason universals must be grounded in a necessary mind. Wouldn't you agree?

No. What equations express are necessarily true but the numbers in the equation do not exist necessarily because the concept of "number" is just a useful thing we invented. The truth of the equation exists independent of a mind but the concepts used to express the equation are mind-conceived and the concepts are not necessary.

I'm talking about universal truths now. Those axioms that are necessarily true, which such arithmetic are predicated upon.

Which I never admitted were mind dependent.
"Music is a zen-like ecstatic state where you become the new man of the future, the Nietzschean merger of Apollo and Dionysus." Ray Manzarek (The Doors)
Apeiron
Posts: 2,446
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1/14/2013 10:02:21 PM
Posted: 3 years ago
At 1/14/2013 9:39:53 PM, phantom wrote:
At 1/14/2013 9:36:08 PM, Apeiron wrote:
At 1/14/2013 9:31:24 PM, phantom wrote:
At 1/14/2013 9:26:22 PM, Apeiron wrote:
At 1/14/2013 8:49:01 PM, phantom wrote:
At 1/14/2013 4:50:41 PM, Apeiron wrote:
At 1/14/2013 4:45:49 PM, phantom wrote:
At 1/14/2013 4:40:48 PM, Apeiron wrote:
At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us. You're correct that they are an invention but why can they not be precise?

Who now, if numbers exist necessarily, then why think they're grounded in contingent minds rather than necessary minds?

Sorry? Numbers don't exist necessarily. It's what they represent that is necessary.

So two apples are necessary?

No. Not what I meant.

Doesn't 2+2=4 true in all possible worlds?

Yes, but numbers don't have to exist in all possible worlds. What the equation represents, holds true to reality necessarily but the concepts of numbers don't except in all worlds because concepts are something agents invent/conceive.

Well that's contradictory, right? If numbers hold in all possible worlds, then they're necessary. But if they're mind-dependant, then they can't be in a contingent mind, unless of course they're contingent. Which isn't what you said, thus by your own reason universals must be grounded in a necessary mind. Wouldn't you agree?

No. What equations express are necessarily true but the numbers in the equation do not exist necessarily because the concept of "number" is just a useful thing we invented. The truth of the equation exists independent of a mind but the concepts used to express the equation are mind-conceived and the concepts are not necessary.

I'm talking about universal truths now. Those axioms that are necessarily true, which such arithmetic are predicated upon.

Which I never admitted were mind dependent.

OK, so we're in agreement then I think.
Sidewalker
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1/15/2013 8:31:54 AM
Posted: 3 years ago
At 1/14/2013 4:19:44 PM, tarkovsky wrote:
I think anyone with more than just a skin deep interest in either philosophy or mathematics might find this an interesting and important question. How do numbers exist? I feel quite comfortable saying that numbers don't have a corporeal existence; I've never encountered in my sensuous world, the number 1 or the constant e. Surely I've observed the physical characteristics of nature, some of which I might be able to describe with a number or some mathematical relation. However, perhaps a more technical and as yet a more demanding question would be how do mathematical objects exist?

My intuition tells me that mathematical objects exist conceptually. Numbers don't exist corporeally and there is no higher mathematical plane of reality from which this physical world derives. In fact, a number is a just a more exact telling of the more general notion of quantity. That said, numbers do not precisely represent the corporeal facts of which the natural world is comprised; all mathematics and numbers can do are stand as an approximation, an invention, a human interpretation of the brute facts of the nature.

For fear of starting off this discussion with what may already be an overly long post I'll stop here.

Let me give you the definitive answer here"there isn"t one.

In general, it"s fair to say that mathematicians just have to live with a peculiar unresolved problem, and that is, what is the nature of mathematical objects such as numbers? Do they exist independently of the human mind or not? When the greatest mathematicians the world ever produced tried to address the problem the result was nothing conclusive. Realistically speaking, there were only three viable candidates, Hilbert"s Formalism, Frege"s Logicism, and Brouwer"s Intuitionism, ony three horses in the race and none of them won.

Most mathematicians will admit that they think mathematical objects exist independently of the mind, that we discover mathematical objects and their properties, we don"t invent them. But then the problem becomes, if mathematical objects exist independently of the mind, then how do we learn about them? We can"t have learned it empirically; mathematical objects don"t exist in the real world.

Are we born knowing these innate things as Descarte"s seemed to believe? Are they "revealed" to us by God as Augustine believed? In the end, the problem of the philosophical foundation is an unresolved problem in Mathematics. A mathematical theory that works, that makes predictions about reality, certainly implies that there is a connection between mathematics and reality. But what is it?

Hilbert and Brouwer said mathematical objects are created by the mind, but on the other side of this, you have Eugene Wigner writing "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" which tells us that there"s got to be more to mathematics than that.

Hilbert"s Formalism fell to Godel"s proof that Formalism isn"t possible by showing that it is provable that you cannot prove the truth of all arithmetic statement within arithmetic, and Turing"s demonstration that you can"t have an effective decision procedure

Frege"s Logicism fell to the self-referential paradox, even after Russell tweaked it with the Principia, the attempt to reduce mathematics to logic failed.

We were are left dangling with Brouwer"s Intuitionism which we intuitively feel cannot be true because the challenge to the law of contradiction left us with nothing to logically work with, and it"s foundational postulate that the truth of a mathematical statement is a subjective claim leaves us with an unacceptable vagueness of the intuitionistic notion of truth. In the end, Brouwer's Intuitionism does nothing to explain "The Unreasonable Effectiveness of Mathematics in the Physical Sciences"

The fact is, the greatest mathematical minds ever produced have pretty much concluded that we do not understand the connection between mathematics and reality.
"It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater
Franz_Reynard
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1/15/2013 9:16:01 AM
Posted: 3 years ago
At 1/15/2013 8:31:54 AM, Sidewalker wrote:

Let me give you the definitive answer here"there isn"t one.

In general, it"s fair to say that mathematicians just have to live with a peculiar unresolved problem, and that is, what is the nature of mathematical objects such as numbers? Do they exist independently of the human mind or not? When the greatest mathematicians the world ever produced tried to address the problem the result was nothing conclusive. Realistically speaking, there were only three viable candidates, Hilbert"s Formalism, Frege"s Logicism, and Brouwer"s Intuitionism, ony three horses in the race and none of them won.

Most mathematicians will admit that they think mathematical objects exist independently of the mind, that we discover mathematical objects and their properties, we don"t invent them. But then the problem becomes, if mathematical objects exist independently of the mind, then how do we learn about them? We can"t have learned it empirically; mathematical objects don"t exist in the real world.

Are we born knowing these innate things as Descarte"s seemed to believe? Are they "revealed" to us by God as Augustine believed? In the end, the problem of the philosophical foundation is an unresolved problem in Mathematics. A mathematical theory that works, that makes predictions about reality, certainly implies that there is a connection between mathematics and reality. But what is it?

Hilbert and Brouwer said mathematical objects are created by the mind, but on the other side of this, you have Eugene Wigner writing "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" which tells us that there"s got to be more to mathematics than that.

Hilbert"s Formalism fell to Godel"s proof that Formalism isn"t possible by showing that it is provable that you cannot prove the truth of all arithmetic statement within arithmetic, and Turing"s demonstration that you can"t have an effective decision procedure

Frege"s Logicism fell to the self-referential paradox, even after Russell tweaked it with the Principia, the attempt to reduce mathematics to logic failed.

We were are left dangling with Brouwer"s Intuitionism which we intuitively feel cannot be true because the challenge to the law of contradiction left us with nothing to logically work with, and it"s foundational postulate that the truth of a mathematical statement is a subjective claim leaves us with an unacceptable vagueness of the intuitionistic notion of truth. In the end, Brouwer's Intuitionism does nothing to explain "The Unreasonable Effectiveness of Mathematics in the Physical Sciences"

The fact is, the greatest mathematical minds ever produced have pretty much concluded that we do not understand the connection between mathematics and reality.

I may not always agree with you, but I sure do enjoy reading your posts.

I wouldn't go so far as to say that man doesn't understand the connection between Mathematics and reality altogether -- but, more that the full extent of humanity is pretty far out of reach of even a mathematician's comprehension.

That is to say, some mathematicians and physicists believe that there is no fabric of reality, and instead, that it could perhaps, with some necessary positive evolution, growth and development of existant concepts, reduce reality to a series of symbols and images that we've produced to analyze and define what we've discovered.

There are others, though, which this post did not take into consideration, that do believe there is a reducible fabric of reality separate of the reducible aggregate of reality (and would be referenced in mathematical statements as yet another symbol), such as, say, Ricci curves, or Planck's equation. Einstein, for example.

It's clear that we've discovered real concepts that apply to reality, which essentially means that our interpretations are real enough to enable us to manipulate the reality we interpret (which answers so many deep philosophical questions, but no one seems to pay attention), but that doesn't mean that the language humans have invented to translate and interpret these concepts would exist despite humanity. It means that these concepts apply to real things that exist despite humanity, sure, but that also necessarily manifested humanity, and thus can only exist, to the extent of our knowledge, with the inclusion of humanity. That, once again, does not prove contingency on humanity for existence, per se, as humanity is more contingent on existence for existence, given humanity is a product of this reality, not the other way around.

Anyway... the point is that our observations are functionally true, meaning that they exist objectively, as a discovery rather than an invented concept. However, the language used to describe these concepts, such as Mathematics, is an invention of humanity, in all of its manifestations. Roman/Arabic numbers are not the only numbers to have ever existed, nor which have been used to accurately describe reality, you know.
Franz_Reynard
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1/15/2013 9:17:08 AM
Posted: 3 years ago
At 1/15/2013 9:16:01 AM, Franz_Reynard wrote:

I wouldn't go so far as to say that man doesn't understand the connection between Mathematics and reality altogether -- but, more that the full extent of reality is pretty far out of reach of even a mathematician's comprehension.

Fixed.
tBoonePickens
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1/15/2013 1:33:42 PM
Posted: 3 years ago
At 1/14/2013 4:19:44 PM, tarkovsky wrote:
I think anyone with more than just a skin deep interest in either philosophy or mathematics might find this an interesting and important question. How do numbers exist? I feel quite comfortable saying that numbers don't have a corporeal existence; I've never encountered in my sensuous world, the number 1 or the constant e. Surely I've observed the physical characteristics of nature, some of which I might be able to describe with a number or some mathematical relation. However, perhaps a more technical and as yet a more demanding question would be how do mathematical objects exist?

My intuition tells me that mathematical objects exist conceptually. Numbers don't exist corporeally and there is no higher mathematical plane of reality from which this physical world derives. In fact, a number is a just a more exact telling of the more general notion of quantity. That said, numbers do not precisely represent the corporeal facts of which the natural world is comprised; all mathematics and numbers can do are stand as an approximation, an invention, a human interpretation of the brute facts of the nature.

For fear of starting off this discussion with what may already be an overly long post I'll stop here.
Of course, this all depends on what you mean by "exist", as someone rightfully pointed out in their post. Exist = possible; numbers are possible ergo they exist.

Added to that, what does root two represent?
The length of the hypotenuse of a right-angle triangle who's other sides are both 1 in length.

Moreover, some mathematical objects are 'discovered' or 'created' (whatever everyone is most comfortable with saying) with respect to the purely theoretical.
They are all more or less discovered but there are some that are the strict consequence of the mathematical system and have no DIRECT corollary to a known physical framework BUT are possible within an ACCEPTED framework (ie mathematics.)

Negative roots were mathematically relevant long before we could determine any worldly analog.
Sure, by accepting certain AXIOMS, then the mathematics allows for a real world analog. Physics = Applied Mathematics.

Same goes for negative numbers.
You can think of negative numbers as the subtraction of positive numbers; they can be thought of as a potential instead of an actual; etc.

*********************************************

At 1/14/2013 4:28:37 PM, phantom wrote:
They exist conceptually because we thought them up, though what they represent in reality exists independent of us.
Well, we thought them up BECAUSE we experienced what they represent in reality.

You're correct that they are an invention but why can they not be precise?
Good question, and I agree with what I think you're alluding to. Numbers can be as precise as we need them to be. For example: if I have one apple in a bag then I have precisely one apple in a bag.

But I think that what the OP probably meant was "accurate" instead of "precise." Precision is the degree of accuracy of a measurement, so the only limiting factor here is technology and the laws of physics.

***********************************

At 1/14/2013 4:32:18 PM, 000ike wrote:
I feel like "exist" is one of those halfway ambiguous, halfway meaningless words. Whenever we claim that anything exists, we need to establish what we mean by it first.
^^^ This.
Exist = Possible.

I am tempted to say "Exist = Physically Possible" but I shall resist and say that this is a distinction without a difference.

Exist can mean that the object occupies physical space, or it can mean for something to occur, or for something to simply be a part of reality.
This means "exist here/now" = "possible here/now".

And claiming that certain things exist can make abundant sense or no sense depending on which meaning is being implied. Consider the question, "do the laws of physics exist?" The answer is yes, if we're referring to existence in the sense of anything concerned with reality - as the "laws of physics" is just a name that describes operations of reality.
Sure, the laws of physics have been observed to be possible.

The same goes for numbers. If existence was used in the sense of the prior definitions, then neither numbers nor laws can exist.
Sure in so much as our empirical observation of numbers: they are certainly possible.

**************************
At 1/15/2013 8:31:54 AM, Sidewalker wrote:

Hilbert's Formalism fell to Godel's proof that Formalism isn't possible by showing that it is provable that you cannot prove the truth of all arithmetic statement within arithmetic...
Any axiomatic system cannot be both Complete and Consistent. Of course, one can always say screw Completeness and just make sure it's consistent! I say Universal Completeness is not possible, at least not the way it's presented by the mathematicians.

...and Turing's demonstration that you can't have an effective decision procedure
Please elaborate...

The fact is, the greatest mathematical minds ever produced have pretty much concluded that we do not understand the connection between mathematics and reality.
Well...I'd say that for the most part mathematics is the OBSERVATION of reality, or at least the mathematics of physics is.
WOS
: At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote:
: Without nothing existing, you couldn't have something.
Sidewalker
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1/15/2013 8:18:39 PM
Posted: 3 years ago
At 1/15/2013 1:33:42 PM, tBoonePickens wrote:
At 1/15/2013 8:31:54 AM, Sidewalker wrote:

Hilbert's Formalism fell to Godel's proof that Formalism isn't possible by showing that it is provable that you cannot prove the truth of all arithmetic statement within arithmetic...
Any axiomatic system cannot be both Complete and Consistent. Of course, one can always say screw Completeness and just make sure it's consistent! I say Universal Completeness is not possible, at least not the way it's presented by the mathematicians.

I"m pretty sure "screw Completeness" is considered an invalid proposition in the philosophy of mathematics.

...and Turing's demonstration that you can't have an effective decision procedure
Please elaborate...

Sure, Hilbert's Formalism program called for the formalization of all of mathematics in axiomatic form and for it to be complete there were three requirements; a formalist system of mathematics must be complete, consistent, and decidable. Godel put a bullet through the first two by proving that any consistent axiomatic system of arithmetic would leave some arithmetical truths unprovable, but this didn"t rule out the possibility of an "effective decision procedure" which could determine whether or not any given proposition was provable or not, which may have saved face for Formalism on a practical basis.

Hilbert's "decision problem" (Entscheidungsproblem in German) said for mathematics to be decidable, there had to be a method which could be applied to any assertion which would produce a correct decision in a finite time on whether the assertion was true or not. Turing proved that Hilbert's decision problem was unsolvable by developing the Turing machine and demonstrating the halting problem, this effectively laid out the foundational basis for the computing machines that we are using to discuss this, and it also put the final bullet through Hilbert's Formalism. Hilbert's Formalism was formally dead, it was proven that any axiomatic theory rich enough to enable the expression and proof of basic arithmetic propositions could be neither complete (Godel) nor effectively decidable (Turing and Church).

BTW, it's my understanding that a couple months before Turing's paper, Alonzo Church had written one proving the decision problem was unsolvable but it was so abstract that nobody realized it till after Turing's paper, plus we didn"t get computers from it, so he rarely gets a mention.


The fact is, the greatest mathematical minds ever produced have pretty much concluded that we do not understand the connection between mathematics and reality.
Well...I'd say that for the most part mathematics is the OBSERVATION of reality, or at least the mathematics of physics is.

I don"t think "for the most part" is considered valid in the philosophy of mathematics either. Mathematics is a complete abstraction that is certainly useful for correlating observations, but the question as to why it is so "unreasonably effective" in science remains unsolved.
"It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater
Sidewalker
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1/16/2013 9:16:39 AM
Posted: 3 years ago
At 1/15/2013 9:16:01 AM, Franz_Reynard wrote:
At 1/15/2013 8:31:54 AM, Sidewalker wrote:

The fact is, the greatest mathematical minds ever produced have pretty much concluded that we do not understand the connection between mathematics and reality.

I may not always agree with you, but I sure do enjoy reading your posts.

Thanks, and same here, the feeling is mutual, but this post might just cure you of that.

I wouldn't go so far as to say that man doesn't understand the connection between Mathematics and reality altogether -- but, more that the full extent of humanity is pretty far out of reach of even a mathematician's comprehension.

Yeah, especially mathematicians, they don"t know dick about humanity, and there"s a half of humanity, an entire gender, that nobody understands.

That is to say, some mathematicians and physicists believe that there is no fabric of reality, and instead, that it could perhaps, with some necessary positive evolution, growth and development of existant concepts, reduce reality to a series of symbols and images that we've produced to analyze and define what we've discovered.

There are others, though, which this post did not take into consideration, that do believe there is a reducible fabric of reality separate of the reducible aggregate of reality (and would be referenced in mathematical statements as yet another symbol), such as, say, Ricci curves, or Planck's equation. Einstein, for example.

I don"t follow what you are trying to say here.

It's clear that we've discovered real concepts that apply to reality, which essentially means that our interpretations are real enough to enable us to manipulate the reality we interpret (which answers so many deep philosophical questions, but no one seems to pay attention), but that doesn't mean that the language humans have invented to translate and interpret these concepts would exist despite humanity. It means that these concepts apply to real things that exist despite humanity, sure, but that also necessarily manifested humanity, and thus can only exist, to the extent of our knowledge, with the inclusion of humanity. That, once again, does not prove contingency on humanity for existence, per se, as humanity is more contingent on existence for existence, given humanity is a product of this reality, not the other way around.

Well, the Idealists think it is the other way around.

Anyway... the point is that our observations are functionally true, meaning that they exist objectively, as a discovery rather than an invented concept. However, the language used to describe these concepts, such as Mathematics, is an invention of humanity, in all of its manifestations. Roman/Arabic numbers are not the only numbers to have ever existed, nor which have been used to accurately describe reality, you know.

I think the idealists have a valid point in terms of making our experiential reality primary. This idea that objective reality comes first, and our thoughts come second might just be an inversion fallacy.

Just given the order by which we know anything, the sequence of cognition by which we come to experience reality, we must accept that it is the existence of mental phenomena that is primary, and the physical word is secondary, it is an inference once removed from what is known directly. The only thing you know directly is your sensations, and from that directly known reality, you presume a cause of those sensations and project it outside of yourself, the existence of a physical reality is an inference, the mental phenomena is known directly and without any mediation of thought, physical reality is only the presumed cause of those sensations. The physical reality that comes to you directly is not substantial, it isn"t concrete, it is an invisible and completely immaterial combination of colorless energy permutations and ephemeral forces arriving at our senses, it is only made concrete by your mind, it exists first and foremost in your mind, you only project its existence "out there" into a concrete physical form, but it all happens "in here".

Therefore, all knowledge of anything outside of mind is mediate, contingent upon some constructive cognitive process projected "out there", our only evidence that there even is a universe, or reality "out there" comes from a "presumption", the existence of a physical reality is inferred at best. By inference we presume that there must be something out there causing these sensations, and then we mentally construct a model of what is out there. What we take as the reality outside of us, outside of consciousness, the so-called objective reality, is a construct that is the "presumed" cause of our sensations. We can only presume that something is "out there" causing these sensations that we are having "in here". The only knowledge that is immediate is "in here", in our consciousness, that is the only thing we know directly. Consequently, all other knowledge is mediated; and everything else is a projection, we only know it by our sensations, reality is the thing we "presume" to be "out there", beyond us; the thing causing our sensations.

So perhaps the inversion mistake is this presumption that the "out there" is first and foremost, and the "in here" is secondary, and we are mistakenly calling the inference real, and calling the only reality we know directly unreal. Objective and subjective are misleading terms, perhaps what is truly subjective is this presumption that what we call objective, out there, is more real than what we know directly in here

There you go, that ought to cure you of this enjoy reading my posts thing :)
"It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater
tBoonePickens
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1/16/2013 5:36:26 PM
Posted: 3 years ago
At 1/15/2013 8:18:39 PM, Sidewalker wrote:
At 1/15/2013 1:33:42 PM, tBoonePickens wrote:
Any axiomatic system cannot be both Complete and Consistent. Of course, one can always say screw Completeness and just make sure it's consistent! I say Universal Completeness is not possible, at least not the way it's presented by the mathematicians.
I"m pretty sure "screw Completeness" is considered an invalid proposition in the philosophy of mathematics.
Why would one have an impossibility as a target? One wouldn't; that's why I said to screw Completeness.

Sure, Hilbert's Formalism program called for the formalization of all of mathematics in axiomatic form and for it to be complete there were three requirements; a formalist system of mathematics must be complete, consistent, and decidable. Godel put a bullet through the first two by proving that any consistent axiomatic system of arithmetic would leave some arithmetical truths unprovable, but this didn"t rule out the possibility of an "effective decision procedure" which could determine whether or not any given proposition was provable or not, which may have saved face for Formalism on a practical basis.
My point is that there is nothing wrong with leaving CERTAIN truths unprovable so long as these are tautologies. It's either this or infinite regression/progression. Certain things need no "proof" because they prove themselves.

Hilbert's "decision problem" (Entscheidungsproblem in German) said for mathematics to be decidable, there had to be a method which could be applied to any assertion which would produce a correct decision in a finite time on whether the assertion was true or not.
Ah yes, I remember. In comp-sci it's kinda like how you cannot write a program/algorithm that determines whether or not another program/algorithm is in an endless loop.

Turing proved that Hilbert's decision problem was unsolvable by developing the Turing machine and demonstrating the halting problem, this effectively laid out the foundational basis for the computing machines that we are using to discuss this, and it also put the final bullet through Hilbert's Formalism. Hilbert's Formalism was formally dead, it was proven that any axiomatic theory rich enough to enable the expression and proof of basic arithmetic propositions could be neither complete (Godel) nor effectively decidable (Turing and Church).
It can be Consistent and Incomplete and Decidable within the areas in which it is complete.

BTW, it's my understanding that a couple months before Turing's paper, Alonzo Church had written one proving the decision problem was unsolvable but it was so abstract that nobody realized it till after Turing's paper, plus we didn"t get computers from it, so he rarely gets a mention.
I see.

The fact is, the greatest mathematical minds ever produced have pretty much concluded that we do not understand the connection between mathematics and reality.
Well...I'd say that for the most part mathematics is the OBSERVATION of reality, or at least the mathematics of physics is.
I don"t think "for the most part" is considered valid in the philosophy of mathematics either. Mathematics is a complete abstraction that is certainly useful for correlating observations, but the question as to why it is so "unreasonably effective" in science remains unsolved.
When mathematics is used in physics it is NOT a complete abstraction; it's quite the opposite! In physics, math is 100% "literal" (ie not abstract); barring "theoretical physics" of course.
WOS
: At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote:
: Without nothing existing, you couldn't have something.
tarkovsky
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1/18/2013 12:57:10 PM
Posted: 3 years ago
It's always interesting to consider the expert's opinions and the who-said-what angle in this discussion. However, I think leaving it at that is a way of stopping short and letting the reputed authorities get away with having all the fun. I say we think for ourselves and skip all the shortcuts.

Sidewalker wrote:
Let me give you the definitive answer here"there isn"t one.

Well I'd say we aren't all in agreement about the ontological structure of mathematics and there isn't any one of us who could put forward a position demonstrating its own truth with absolute certainty. At the same time, the position you are arguing from with this statement seems to be very out of the character of the rest of your post. Just because we can't give a definitive answer doesn't mean that there is no definitive answer.

Franz_Reynard wrote:
Anyway... the point is that our observations are functionally true, meaning that they :exist objectively, as a discovery rather than an invented concept. However, the :language used to describe these concepts, such as Mathematics, is an invention of :humanity, in all of its manifestations. Roman/Arabic numbers are not the only :numbers to have ever existed, nor which have been used to accurately describe :reality, you know.

Had to reread this and the preceding paragraph more than once to really figure out what you were getting at, but I think it's a good point. The concepts which we articulate through our mathematics are true, but the math itself is merely the stepladder, and that there is truth beyond symbolic truth.

The subtle point, though, I think we need to make is that only some of the math is true in this way. That is, only those mathematical disciplines, founded on particular axioms, and that we can find worldly meaning and applications to are objectively true in this sense.
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1/19/2013 6:58:20 AM
Posted: 3 years ago
At 1/18/2013 12:57:10 PM, tarkovsky wrote:
It's always interesting to consider the expert's opinions and the who-said-what angle in this discussion. However, I think leaving it at that is a way of stopping short and letting the reputed authorities get away with having all the fun. I say we think for ourselves and skip all the shortcuts.

Sidewalker wrote:
Let me give you the definitive answer here"there isn"t one.

Well I'd say we aren't all in agreement about the ontological structure of mathematics and there isn't any one of us who could put forward a position demonstrating its own truth with absolute certainty. At the same time, the position you are arguing from with this statement seems to be very out of the character of the rest of your post. Just because we can't give a definitive answer doesn't mean that there is no definitive answer.

I concur; I was just giving a brief history of the philosophical debate among professional mathematicians to point out the fact that there isn"t a generally accepted solution to the subject of this thread. I say we think for ourselves too, but I doubt any of us are going to mathematically out think people like G"del and Hilbert and it would certainly be prudent to avoid going down any roads that have already been shown to be dead ends.

Perhaps thinking for ourselves is a matter of what we choose to think about, and how we choose to think.

When we are presented with a logically unsolvable problem it probably means we are framing the problem all wrong. I think the solution lies in unpacking the dynamic between subjective and objective that lies at the heart of most of these unresolved philosophical problems. It seems to me that every postulated solution comes down to choosing between one of two polar opposites, it"s either a matter of deducing from verifiable outer realities to unverifiable inner states, or inferring from verifiable inner states to unverifiable outer realities.

I think we should question whether we need to divide "inner" and "outer" in this way at all. Perhaps we need to rethink our general interpretations of human experience and recognize that such polar opposites as "inner" and "outer" are in fact inseparable opposites, they are mutually sustaining aspects that together constitute a whole.

Descartes is the one who told us to think of them as basically separate from each other, I say we let him figure out how to put them back together. I think the solution is to reject his bifurcation of reality into such oppositions and just accept that the human experience is real. We can simply recognize that "inner" and "outer" are two reciprocal aspects of the singular human experience of reality, neither is more real than the other, they reference two sides of one and the same coin so to speak.

Perhaps the answer lies in overcoming these intellectual detachments of oppositions, coming to terms what it means to be fully human, and making a commitment to that discernment. Life is a gift; each of us has the opportunity to live life to the fullest and to be free of such detachments" to celebrate the gift. In the end analysis, what is most real is the uniquely human experience of a many-leveled, multifaceted reality that includes both "inner" and "outer" united into a synthesis of mind, body and soul, we can choose to be a fully engaged agent that is both a part of reality and is participating in reality.
"It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater