They are terms to describe a specific quantity. 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.
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
---------- Post added 05-19-2010 at 11:04 AM ----------
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.
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".
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
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.
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.