What is the real meaning of natural numbers?

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What is the real meaning of natural numbers?
Superficially they can be used in counting and other mathematical purposes. But when examined more deeply,
they exhibit relationships that make me wonder just how interrelated sets of natural numbers really are.
One of my questions is what is the true purpose of numbers? Is it for mundane things such as counting or multiplying? How about recreational purposes such as puzzles or math objects such as magic squares? Is there a limit to the potential of numbers?
My opinion is there is no limit to what you can do with numbers, whether it be for practical applications or recreational purposes. Have you given any thought to this?
PhilX
they exhibit relationships that make me wonder just how interrelated sets of natural numbers really are.
One of my questions is what is the true purpose of numbers? Is it for mundane things such as counting or multiplying? How about recreational purposes such as puzzles or math objects such as magic squares? Is there a limit to the potential of numbers?
My opinion is there is no limit to what you can do with numbers, whether it be for practical applications or recreational purposes. Have you given any thought to this?
PhilX
Re: What is the real meaning of natural numbers?
I see mathematics as a language, a means of describing the world, more detailed and accurate than other languages  (arguably) in order of sophistication  smell, body language, spoken words and written words.
Each increase in the ability of a language to accurately describe reality resulted in greater empowerment  smelling, seeing and hearing, speaking, writing, calculating.
Each increase in the ability of a language to accurately describe reality resulted in greater empowerment  smelling, seeing and hearing, speaking, writing, calculating.

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Re: What is the real meaning of natural numbers?
Your question asked about the limits of natural numbers, and I believe your conclusion is that they are not limited. Yet, mathematics tells us that natural numbers are very limited, which is why we developed integers, rational numbers, real numbers, and finally, complex numbers. One cannot even perform many operations in subtraction if one is solely limited to natural numbers. New numbers were developed precisely because of limitations that were discovered with the existing numbers being used.

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Re: What is the real meaning of natural numbers?
Without the concept of natural numbers, you wouldn't have any of those "new numbers."Science Fan wrote: ↑Mon Jun 12, 2017 5:38 pmYour question asked about the limits of natural numbers, and I believe your conclusion is that they are not limited. Yet, mathematics tells us that natural numbers are very limited, which is why we developed integers, rational numbers, real numbers, and finally, complex numbers. One cannot even perform many operations in subtraction if one is solely limited to natural numbers. New numbers were developed precisely because of limitations that were discovered with the existing numbers being used.
PhilX
 Arising_uk
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Re: What is the real meaning of natural numbers?
Apparently you think there is a real meaning so can you state it first please?

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Re: What is the real meaning of natural numbers?
You shouldn't make an assumption like that about me as the body of knowledge concerning natural numbers keeps growing. It started off simply and has evolved over the years with discoveries that keep going on. The most basic question about natural numbers is whether they really exist or just in our minds?Arising_uk wrote: ↑Mon Jun 12, 2017 11:47 pmApparently you think there is a real meaning so can you state it first please?
Let me ask you this. What do natural numbers mean to you? Do you think they're the most important type of number or do you have a greater interest in other types of numbers? Do you even know what a natural number is? What hidden properties do natural numbers have that await discovery?
PhilX
 Arising_uk
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Re: What is the real meaning of natural numbers?
Then why did you say "What is the real meaning..."?Philosophy Explorer wrote:You shouldn't make an assumption like that about me as the body of knowledge concerning natural numbers keeps growing. ...
What on earth do you mean by "really exist" with respect to numbers? At base the natural numbers were created as symbols for counting objects and all they really are is shorthand for '1' and lots of '1''s.It started off simply and has evolved over the years with discoveries that keep going on. The most basic question about natural numbers is whether they really exist or just in our minds? ...
They mean that I can count or describe the number of things easier.Let me ask you this. What do natural numbers mean to you? ...
How do even rank numbers by 'importance'?Do you think they're the most important type of number or do you have a greater interest in other types of numbers? ...
Sure, the integers.Do you even know what a natural number is? ...
I doubt they have any hidden properties as they are just shorthand for lots of '1's. Are there relationships between the symbols due to the existence of the mathematical operators that we haven't found yet in the axiomatic system for the integers, I guess so but you'd have to ask a Mathematician as Logic is just about my game not Maths.What hidden properties do natural numbers have that await discovery?

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Re: What is the real meaning of natural numbers?
I will answer in reverse order.Arising_uk wrote: ↑Tue Jun 13, 2017 2:15 amThen why did you say "What is the real meaning..."?Philosophy Explorer wrote:You shouldn't make an assumption like that about me as the body of knowledge concerning natural numbers keeps growing. ...What on earth do you mean by "really exist" with respect to numbers? At base the natural numbers were created as symbols for counting objects and all they really are is shorthand for '1' and lots of '1''s.It started off simply and has evolved over the years with discoveries that keep going on. The most basic question about natural numbers is whether they really exist or just in our minds? ...They mean that I can count or describe the number of things easier.Let me ask you this. What do natural numbers mean to you? ...How do even rank numbers by 'importance'?Do you think they're the most important type of number or do you have a greater interest in other types of numbers? ...Sure, the integers.Do you even know what a natural number is? ...I doubt they have any hidden properties as they are just shorthand for lots of '1's. Are there relationships between the symbols due to the existence of the mathematical operators that we haven't found yet in the axiomatic system for the integers, I guess so but you'd have to ask a Mathematician as Logic is just about my game not Maths.What hidden properties do natural numbers have that await discovery?
"I doubt they have any hidden properties as they are just shorthand for lots of '1's. Are there relationships between the symbols due to the existence of the mathematical operators that we haven't found yet in the axiomatic system for the integers, I guess so but you'd have to ask a Mathematician as Logic is just about my game not Maths."
I could ask what you're doing on this thread since logic is your game but not math, but since you decided to intrude, I'll deal with it. I presume by"...just shorthand for lots of '1's", you mean something like 4 = 1 + 1 + 1 + 1 e.g. How about 2 x 2 = 4? Not quite the shorthand you envisioned,
is it? Then you said "...axiomatic system for the integers." This "system" isn't taught in college (there is an axiomatic system for the real numbers which is a different story) so you won't find any mathematical operators there and logic which you said is your game won't help you out here either.
Do you even know what a natural number is? ... "Sure, the integers." Really!!!? What does your logic say about 1? That's an integer, but not a natural number. I suggest you refresh your memory about the difference between integers and natural numbers by looking up their definitions because I know you "just" won't take my word for it.
"How do [we] even rank numbers by 'importance'?" Well, for one thing, different types of numbers arose throughout history which allowed mathematicians and scientists to make progress in many different ways in their investigations of phenomena. Also importance can be a personal preference. For example an accountant would look at numbers differently than a Sudoku puzzle lover. For you, you said you can count or describe the number of things easier.
"What on earth do you mean by 'really exist' with respect to numbers? At base the natural numbers were created as symbols for counting objects..." As I had asked, do they just exist inside of your mind or do they have physical existence?
"Then why did you say 'What is the real meaning...'?"
As more discoveries are made about how the natural numbers interact among themselves, the more interesting they get as they show unusual properties and discoveries are made all the time about them in numerous ways. Under these circumstances, it would be natural to ask what is the real meaning of natural numbers.
PhilX

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Re: What is the real meaning of natural numbers?
Remember that Maths is a subset of LogicArising uk wrote:
Logic is just about my game not Maths

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Re: What is the real meaning of natural numbers?
Two things:surreptitious57 wrote: ↑Tue Jun 13, 2017 6:22 amRemember that Maths is a subset of LogicArising uk wrote:
Logic is just about my game not Maths
1) Math logic is a subset of logic
2) If maths were a subset of logic (I'm not saying I agree to this), then why is this category titled to show both maths and logic instead of just logic?
PhilX
 Arising_uk
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Re: What is the real meaning of natural numbers?
I thought Russell, Whitehead and Godel showed this not the case?surreptitious57 wrote:Remember that Maths is a subset of Logic
p.s.
Unless of course by Logic you mean the study of formal axiomatic systems?
 Arising_uk
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Re: What is the real meaning of natural numbers?
Well given that the philosophy of maths and metamathematics very much involves mathematical logic and philosophy I hardly see it as an intrusion and not least because this is an open forum.Philosophy Explorer wrote:I will answer in reverse order.
I could ask what you're doing on this thread since logic is your game but not math, but since you decided to intrude, I'll deal with it. ...
Pretty much is as 2 x 2 is shorthand for adding the group two to itself twice, 2+2, and at base this is 1+1+1+1.I presume by"...just shorthand for lots of '1's", you mean something like 4 = 1 + 1 + 1 + 1 e.g. How about 2 x 2 = 4? Not quite the shorthand you envisioned, is it? ...
Apparently they don't do mathematical logic nor philosophy of mathematics nor the history of mathematics at the college level then(this would explain your ridiculous question) as axiomatic systems for the positive integers have been invented since the C19th  Peano's being a most notable one.Then you said "...axiomatic system for the integers." This "system" isn't taught in college (there is an axiomatic system for the real numbers which is a different story) so you won't find any mathematical operators there and logic which you said is your game won't help you out here either. ...
My apologies I should have said the positive integers.Do you even know what a natural number is? ... "Sure, the integers." Really!!!? What does your logic say about 1? That's an integer, but not a natural number. I suggest you refresh your memory about the difference between integers and natural numbers by looking up their definitions because I know you "just" won't take my word for it. ...
Which different types of numbers?Well, for one thing, different types of numbers arose throughout history which allowed mathematicians and scientists to make progress in many different ways in their investigations of phenomena. Also importance can be a personal preference. For example an accountant would look at numbers differently than a Sudoku puzzle lover. For you, you said you can count or describe the number of things easier.
Depends what you mean? This 1 has a physical existence upon this screen but are you talking about a perfect platonic "1" living in some ideal world of forms? If so I'd say not, they are in our minds and they arise from there being objects to count.As I had asked, do they just exist inside of your mind or do they have physical existence?
Since they are a formal axiomatic system I'd have thought any properties they show are the consequence of the operators and proof method in use? Any question of 'real meaning' seems to be like asking "What is the real meaning of pink?". You seem to view numbers in more of a kabbalahistic or occult manner in that they can reveal some mysterious truth about reality?As more discoveries are made about how the natural numbers interact among themselves, the more interesting they get as they show unusual properties and discoveries are made all the time about them in numerous ways. Under these circumstances, it would be natural to ask what is the real meaning of natural numbers.

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Re: What is the real meaning of natural numbers?
"(this would explain your ridiculous question)" Ridiculous just to you so far, another indication of your intrusion.Arising_uk wrote: ↑Tue Jun 13, 2017 12:19 pmWell given that the philosophy of maths and metamathematics very much involves mathematical logic and philosophy I hardly see it as an intrusion and not least because this is an open forum.Philosophy Explorer wrote:I will answer in reverse order.
I could ask what you're doing on this thread since logic is your game but not math, but since you decided to intrude, I'll deal with it. ...Pretty much is as 2 x 2 is shorthand for adding the group two to itself twice, 2+2, and at base this is 1+1+1+1.I presume by"...just shorthand for lots of '1's", you mean something like 4 = 1 + 1 + 1 + 1 e.g. How about 2 x 2 = 4? Not quite the shorthand you envisioned, is it? ...Apparently they don't do mathematical logic nor philosophy of mathematics nor the history of mathematics at the college level then(this would explain your ridiculous question) as axiomatic systems for the positive integers have been invented since the C19th  Peano's being a most notable one.Then you said "...axiomatic system for the integers." This "system" isn't taught in college (there is an axiomatic system for the real numbers which is a different story) so you won't find any mathematical operators there and logic which you said is your game won't help you out here either. ...My apologies I should have said the positive integers.Do you even know what a natural number is? ... "Sure, the integers." Really!!!? What does your logic say about 1? That's an integer, but not a natural number. I suggest you refresh your memory about the difference between integers and natural numbers by looking up their definitions because I know you "just" won't take my word for it. ...Which different types of numbers?Well, for one thing, different types of numbers arose throughout history which allowed mathematicians and scientists to make progress in many different ways in their investigations of phenomena. Also importance can be a personal preference. For example an accountant would look at numbers differently than a Sudoku puzzle lover. For you, you said you can count or describe the number of things easier.Depends what you mean? This 1 has a physical existence upon this screen but are you talking about a perfect platonic "1" living in some ideal world of forms? If so I'd say not, they are in our minds and they arise from there being objects to count.As I had asked, do they just exist inside of your mind or do they have physical existence?Since they are a formal axiomatic system I'd have thought any properties they show are the consequence of the operators and proof method in use? Any question of 'real meaning' seems to be like asking "What is the real meaning of pink?". You seem to view numbers in more of a kabbalahistic or occult manner in that they can reveal some mysterious truth about reality?As more discoveries are made about how the natural numbers interact among themselves, the more interesting they get as they show unusual properties and discoveries are made all the time about them in numerous ways. Under these circumstances, it would be natural to ask what is the real meaning of natural numbers.
Which different types of numbers?Also importance can be a personal preference. For example an accountant would look at numbers differently than a Sudoku puzzle lover. For you, you said you can count or describe the number of things easier.
The same set of natural numbers, but under different formats. Accountants tend to look at long columns of numbers which can fall under debits and credits while Sudoku puzzles are built upon 3 x 3 magic squares which again involve natural numbers. So importance depends in part on who you are and what use you put the numbers to.
"Since they are a formal axiomatic system I'd have thought any properties they show are the consequence of the operators and proof method in use? Any question of 'real meaning' seems to be like asking "What is the real meaning of pink?". You seem to view numbers in more of a kabbalahistic or occult manner in that they can reveal some mysterious truth about reality?"
I don't and I never made a claim that they can reveal the truth about reality. I'm just seeking the real meaning about natural numbers.
PhilX
 Arising_uk
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Re: What is the real meaning of natural numbers?
And like I said, ridiculous question as what point the "real" unless you think there is some hidden or unique meaning to the numbers other than just being the product of the need to count efficiently and that we are symbol users?
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