Reply Tue 18 May, 2010 03:34 am
What are numbers ? In what way do they exist? Perhaps we should concentrate on the natural numbers. Personally, I think the natural numbers are based on an iterated intuition of indeterminate unity. Others say that numbers would be here even if we weren't. Are numbers discovered, invented, or both? Or something else?
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Khethil
 
  1  
Reply Tue 18 May, 2010 06:12 am
@Reconstructo,
Reconstructo;165669 wrote:
What are numbers ?


Representations of quantity

Reconstructo;165669 wrote:
In what way do they exist?


They are terms to describe a specific quantity. As such, they're only conceptional and have no existence past what they represent.

Reconstructo;165669 wrote:
Others say that numbers would be here even if we weren't. Are numbers discovered, invented, or both? Or something else?


They're not "here", they're just concepts whose only meaning is inside the mind of the perceiver. Mathematics is a system of representations and concepts that correlate to what we could do if we had "X" man of <whatever>. It allows us to manipulate these, on paper (or in the mind) without having to do it physically to come up with solutions, postulates and hypotheticals.

The universe would continue to operate if we couldn't describe or conceive of it; whether that thing over there has a name has no bearing on its existence, behavior or characteristics - only in the way I perceive, conceptualize or describe it. Its the same with numbers; if it takes 200 rocks to stop the flow of this creek, the number contained is only my way of describing it. Nothing more.
xris
 
  1  
Reply Tue 18 May, 2010 07:47 am
@Khethil,
Two is one more than one and you can share two but you cant one. So man has a reason to debate and dispute the most simplest of concepts.
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Reconstructo
 
  1  
Reply Tue 18 May, 2010 09:11 pm
@Khethil,
Khethil;165685 wrote:
Representations of quantity

I agree. What is quantity?

---------- Post added 05-18-2010 at 10:12 PM ----------

Khethil;165685 wrote:

They are terms to describe a specific quantity. As such, they're only conceptional and have no existence past what they represent.

I agree with this too. But how did we get them? I have my theories, but what are yours?

---------- Post added 05-18-2010 at 10:13 PM ----------

Khethil;165685 wrote:
They're not "here", they're just concepts whose only meaning is inside the mind of the perceiver.


I can dig it. But what is this "mind of the perceiver"? Is it any more real than numbers are?

---------- Post added 05-18-2010 at 10:16 PM ----------

xris;165714 wrote:
Two is one more than one and you can share two but you cant one. So man has a reason to debate and dispute the most simplest of concepts.


The simplest concepts are what the big concepts are made of. Perhaps you think scientists are fools for looking at quarks.
"Two is one more than one"? Wake up! What is this "one"? "Oh well, uh....one is ...uh....well, one!"

---------- Post added 05-18-2010 at 10:17 PM ----------

Khethil;165685 wrote:

The universe would continue to operate if we couldn't describe or conceive of it; whether that thing over there has a name has no bearing on its existence, behavior or characteristics - only in the way I perceive, conceptualize or describe it. Its the same with numbers; if it takes 200 rocks to stop the flow of this creek, the number contained is only my way of describing it. Nothing more.


From a pragmatic perspective I agree. But it should be acknowledged, perhaps, that the universe devoid of man is one of man's abstractions and nothing he has experienced directly.
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jeeprs
 
  1  
Reply Wed 19 May, 2010 04:56 am
@Khethil,
Khethil;165685 wrote:
As such, they're only conceptional and have no existence past what they represent.


The only slightly inconvenient argument against that idea is that mathematics has the uncanny ability to predict things which nobody has ever thought of, let alone represented. There are thousands of examples of this. For example Einstein was able to predict the existence and behavior of bodies that had never been seen and could not be verified for decades afterwards. So the meaning of numbers extends well beyond one potato-two potato three-potato etc.

The other, related, problem is that of abstract proofs and the like. Mathematicians - incidentally I am not one of them - come up with all kinds of abstractions which are not represented by anything in the real world. And then, dammit, some other mathematician will come along and show them wrong, again without reference to anything other than number. So you and I can't go along and find out which of the two is right with reference to anything in reality (or anything existing, anyway.)

So I am afraid naive representationalism does not get you very far. The correspondence theory of truth has many problems like this, as I have learned on the Forum.
Khethil
 
  1  
Reply Wed 19 May, 2010 09:43 am
@jeeprs,
Khethil;165685 wrote:
They are terms to describe a specific quantity. As such, they're only conceptional and have no existence past what they represent.
jeeprs;166047 wrote:
The only slightly inconvenient argument against that idea is that mathematics has the uncanny ability to predict things which nobody has ever thought of, let alone represented. There are thousands of examples of this. For example Einstein was able to predict the existence and behavior of bodies that had never been seen and could not be verified for decades afterwards. So the meaning of numbers extends well beyond one potato-two potato three-potato etc.


We're talking about two distinctly different - yet obviously related - concepts.

You're right in that theorems do allow us to 'sense' or predict many phenomena; and yes, I'm aware of many that have been confirmed long after the formulae first predicted. But I'm talking about numbers alone, not mathematics or formulae. Yes, it makes a difference.

Because the concept of numbers (strictly 'quantity', how many of <yada>), at its basic level, has been defined to correlate directly to observables, it - by definition - can obviously be used to, quite accurately, postulate and calculate a litany of phenomena. This doesn't mean that number's aren't concepts only, it simply means that numerical representations can be used to describe or learn about objects and other phenomena that do have corporeal form. Because I can illustrate observable behaviors for the concept of 'affection' doesn't mean affection isn't just an idea or generalized descriptor (this isn't a direct example or illustration that can be compared, its given only to punctuate how concepts that have no objective form outside the brain can and do often correlate, in their use, with that which lies outside the brain). I hope I've describe this sufficiently... I"m not sure I've done my point justice. Here's hoping it comes across well enough.

The idea of Numbers is still just a concept, regardless of instances of accurate use.

Thanks

---------- Post added 05-19-2010 at 11:04 AM ----------

Reconstructo;165939 wrote:
I agree. What is quantity?
The way I see it, 'quantity' is a broad term to describe a number we've given to precisely describe groups of repeating (or similar) iterations in whatever it is we're talking about.

Reconstructo;165939 wrote:
I agree with this too. But how did we get them? I have my theories, but what are yours?


I think we developed them to describe more accurately objects and occurrences in our world. The emergence of numbers throughout history in the various languages, to me, is no more a surprise than us coming up with words to describe "high", "low" or "dog".

Reconstructo;165939 wrote:
I can dig it. But what is this "mind of the perceiver"? Is it any more real than numbers are?
Yes, I think so. Because we can't hold in our hands consciousness doesn't mean it's not a phenomena that doesn't exist while numbers are - by definition - simply representations. As far as consciousness goes: Despite what we think it is, might be, how it works, etc., we have simply too much evidence in everyday life that this thing we call "consciousness" - by definition - is a process that actually takes place. Conversely, numbers are - by definition - representations of quantity; just representations.

In this context, the mind of the perceiver (in which the concept of numbers lies) would refer to the mental processes (primarily biochemical and bio-electrical) that constitute the ability for a person to grasp, define and demonstrate the concept.

Reconstructo;165939 wrote:
The simplest concepts are what the big concepts are made of. Perhaps you think scientists are fools for looking at quarks. "Two is one more than one"? Wake up! What is this "one"? "Oh well, uh....one is ...uh....well, one!"
I'm not quite sure where you're going with this. One shouldn't go to the extremity of saying that because numbers are just a concept, that they therefore have no correlation to 'reality' or worth - quite the contrary!

Thanks much - I hope this makes sense.
Reconstructo
 
  1  
Reply Thu 20 May, 2010 04:35 pm
@Khethil,
Khethil;166143 wrote:

---------- Post added 05-19-2010 at 11:04 AM ----------

The way I see it, 'quantity' is a broad term to describe a number we've given to precisely describe groups of repeating (or similar) iterations in whatever it is we're talking about.

I agree. I think the number system is generated from iterations of an intuited unity. From my reading, matching seems to have developed before counting. For instance, sheep let out to pasture were each associated with a rock, and when the sheep returned, every sheep could be matched with the saved rocks..to see if any were missing.

So the mind has already abstracted the particular sheep from its environment as a particular sheep, as a thing, as a unity. The mind has divided sheep from not-sheep. Unless the mind can divide experience into particular objects, counting is not possible, or really even conceivable.

I think it's safe to assume that many animals can do this, by simply looking at their behavior. And perhaps it's so obvious as to be for most not interesting. But it fascinates me because I am interested in the relationship between our understanding of the discrete and the continuous. I feel that unity is the atom not only of math but also of language, as our nouns refer to unities. See what I mean? I actually got into math after thinking about Kant and Heidegger. The transcendental unity of apperception and the Being of beings.

---------- Post added 05-20-2010 at 05:41 PM ----------

Khethil;166143 wrote:

I think we developed them to describe more accurately objects and occurrences in our world. The emergence of numbers throughout history in the various languages, to me, is no more a surprise than us coming up with words to describe "high", "low" or "dog".

Well it's not in itself any more of a surprise than the rest of our culture. On the other hand, I think all life is a surprise when one is in a certain state of mind. And it's often a good state of mind.

Let's say that we causally connect our experiences. This explains them in one way but not another. Because where did this entire causally understood system come from in the first place? I know Kant's objections to such questions but it's almost an attack on wonder to try to reduce "why is there something rather than nothing" to the sort of question that wants a causal answer. The question is more of a lyrical expression.

Kant's categories fascinate me. I feel that "unity" is the fundamental category from which the others can largely be derived. However this interest of mind is hardly universal or necessary. Smile

---------- Post added 05-20-2010 at 05:43 PM ----------

Khethil;166143 wrote:

I'm not quite sure where you're going with this. One shouldn't go to the extremity of saying that because numbers are just a concept, that they therefore have no correlation to 'reality' or worth - quite the contrary!

Thanks much - I hope this makes sense.


Ah, but when did I say such a thing? I think numbers are not only extreme useful, but aesthetically pleasing in a deep way. Numbers considered apart from associated material objects are something like sculpture. I love those famous equations for pi as an infinite series. Of course pi is a "high tech" transcendental number but I also find 1 and 0 more than a little resonant. So did Leibniz, who seems to have given the West its binary base. (As you may know this goes at least as far back as the I Ching in the East..)

---------- Post added 05-20-2010 at 05:50 PM ----------

jeeprs;166047 wrote:
The only slightly inconvenient argument against that idea is that mathematics has the uncanny ability to predict things which nobody has ever thought of, let alone represented.

An interesting point.

I've noticed that many physics equations are about proportion. For instance, one factor can be inversely proportional to another. But this relationship is possibly continuous. And we yet we cannot write quantities as continuities. So perhaps nature is largely continuous. And our discrete iterations are used to measure and describe this continuity as well as they can.

Of course some of physics is eerily discrete. In this case, the numbers seem to match perfectly. It's a tricky issue. I'm not physicist or mathematician enough to pretend authority on the matter. But the proportional nature of many of our equations is fascinating. Maybe on a macro level, continuous proportion (which can be represent for the most part with continuous variables --"F = ma") is the way we experience reality. And maybe when we zoom in, we start to see particles, particles, particles.
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Reconstructo
 
  1  
Reply Mon 24 May, 2010 01:00 am
@Reconstructo,
I just wanted to add what seems to me like the impossibility of ideal or perfect measurement. Can we not imagine infinite precision? What does this mean for us?
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Reconstructo
 
  1  
Reply Wed 2 Jun, 2010 12:51 am
@Reconstructo,
n is the abstraction of an abstraction, the variable number.....
0 Replies
 
 

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