Malik, thanks for your post.
here is a great link that has somethying to say about our topic, whether numbers really exist or not. please take a look at :
www.math.hawaii.edu/~lee/exist.html. basically, it is very interesting. it says that numbers and math are indeed inventions of humankind and that numbers have a type of
objective existance, meaning they rely on an observer's point of view in order to exist (ie, a Human. sound familiar??). My whole argument has been that we humans invented math and numbers and that without us, they just do not exist. If you read on, it details various research and philisophies on the subject. You may take a different view of the results presented, please let me know.
I know it doesn't address our discussion on actual structure. Her eis a blurb.
The dispute over the existence (or reality) of mathematical entities is an example of what philosophers call the Problem of Universals, something which goes back as far as Plato. This is the question as to whether abstract concepts have some sort of real existence in the world, or whether they exist only in our minds. Like most philosophical problems, it seems to be more a question about language than a question about the world, although there are certainly philophers who would disagree with me in this respect.
I believe it was Kronecker who said, "The natural numbers were created by God; all the others are the invention of humans." I believe that most contemporary mathematicians would agree that Kronecker was wrong only in his statement about natural numbers; they too are the creation of human minds.
Certainly numbers do not have a tangible existence in the world. They exist in our collective consciousness. And yet they are not arbitrary products of our imaginations in the way that fictional characters are.
For instance, when a mathematician says that there exists a prime number which is the sum of two squares, his statement is not a product of his imagination. It is not a matter of opinion. The prime number 13, in fact, is the sum of 3 squared and 2 squared: 13 = 9 + 4. And when the Indian mathematician Ramanujan said to his fellow mathematician G.H. Hardy that 1729 is the smallest number that can be written as a sum of two cubes in two different ways, he was making a statement of fact:
1729 = 10³ + 9³ = 12³ + 1³.
The fact that no smaller number can be so written can be verified, with the help of a computer program or spreadsheet, by listing the values of m³ + n³ for m and n between 1 and 12 and seeing that there are no duplications smaller than 1729 in the list. (One can also note that if m³ + n³ is 1729, then one of these two numbers must be larger than 9 and, of course, no larger than 12. This leaves us with only a few possibilities to check.)
"There exists" is one of the most common phrases in mathematical discourse, and whether one is talking about a prime number with a certain property or the solution to a differential equation, a statement about existence is a statement of fact, not a matter arbitrary choice or opinion.
So numbers do have some sort of objective existence, even though not a tangible existence.
We may have invented numbers. But we did NOT invent their structure. If we invented their structure, then we wouldn't be surprised by their properties. For instance, when we wrote down the formulas for fractals, we didn't suspect the infinitely complex structure these formulas described. We found that out once we plugged them into computers and plotted out that structure visually. Clearly, the fractal formulas contained vast amounts of structure which we didn't put into them, nor did we suspect it was there in the first place. In no sense of the word did we "invent" that structure, we discovered it--much like we discovered the fact that the square of the hypotenuse is equal to the squares of the sides. We didn't invent this relationship. It could not have been otherwise.
Agree we didnt invent the relationship --it is part of the universe! I have never said different. What I say is that we
did invented the math that describes and quantifies it. when we quantify something, we are ordering it.
If we impose order it, rather than discover it, then why can't we pick whatever formula we want to describe it? Why are some formulas accurate, while others are not?
Earth attracts a meteor. why did the meteor fall to earth? because at some point the gravitational pull of the earth was strong enough to overcome the forces acting on the meteor that would cause it to move away from the earth. These are all relationships that occur in nature.
Now lets ask questions: How much gravitational force does the earth affect on the meteor? how fast was the meteor moving in relation to the earth? how fast will it fall? How big a hole will it make when it hits?
I cannot answer these in universal terms, only human ones. So I invent a language and symbols to do so. How am I not imposing order on the force when I define a unit to measure it in, define a standard to go by, and assigning a quantity based on my measurement? Was any of this here before I did it? nope. There was only a force. there is still onty a force. but in my mind I can now think about it becasue I have ordered it in my head.
Now by
imposing I do not mean I am changing or affecting the force; the order I am imposing is in my own perception of it -- the force itself has no order for me until I quantify it using terms I understand.
Obviously, there is MUCH more going on than mere definition and imposing order on the system. We are discovering something that is already there, and finding which formula accurately matches it. That "something" has a structure which precludes every single formula . . . except the right one. The one that accurately describes it. If it's completely up to us, then why are some formulas right, while others are not?
I think we agree here but that we define structure differently. I agree their is only one formula that can describe a relationship. I agree the relationship is part of nature. But my point is our math merely describes and quantifies the relationship using completely human concepts. I will argue that math itself is inexact -- we can calculate gravitational force, but only so far as our instruments are accurate. there is a limit to how accurate our measurements can be. So math is in most cases an approximation, however accurate that approximation is.
Dude, you don't seriously think I'm confusing units we invent with structure inherent in the world, do you? Have I given you any evidence that I'm that stupid? I'm not saying that seconds and meters exist beyond our definitions. For you to suggest this implies a fundamental misunderstanding with my position. It's no wonder we're making no progress. The inverse square law is true no matter which invented units we use. Why do you think that is??? It's because there is a truth here which transcends invented symbols.
You may have issues, but being stupid is not one of them. I agree inverse square law is true anywhere and is possible with any quantity and unit. but those quantities and units are concepts invented by humans. yes there is a truth there (the relationship), and we can
describe that truth using math. But the truth is not quantified in terms we understand, so we invent a symbolic representation of it that we can understand.
So, my question--which you still have not answered--is why does math describe this form accurately?Indeed, how can the question have meaning at all if mathematical descriptions are completely up to us? "Accuracy" can only have meaning in this context if there is something about the system which is correctly described by the formula. The issue of accuracy is meaningless if math is nothing more than imposing order, and defining terms.
Sigh. Of course the relationship is described accurately by our math , or accurately enough for us. We can never express Pi as a completely accurate number, no matter how hard we try.
What I am saying is that we have invented a language to describe a relationship or property of some force, etc. we observe in nature. Why does us inventing it preclude it being accurate?
