quote:Originally posted by MBM:quote:Originally posted by HonestBrother:
I specialize in Number Theory.
From a universal perspective, aren't numbers really pretty much concepts without meaning?
No more than 'justice', 'freedom', or 'beauty'.
Since numbers are infinite, isn't the difference between any two numbers meaningless?
It depends upon the point of view that one adopts. I'll return to this question when I have more time and my head is clear.
If numbers are supposed to attach value and describe difference, then in the realm of theory when we consider all numbers, how do we differentiate one from another? It seems like their values are all "fungible" if there are an infinite number of numbers on either side of them on a number line.
There are a number of ways one can classify numbers and make real distinctions between them. For example, some whole numbers are prime numbers - like 2, 3, 5, 7, 11, 13, 17, 19, etc. - while other numbers are composite - like 4, 6, 8, 9, 10, 12, 14, etc.
The numbers in the first group can be expressed as a product of two whole numbers in one and only one way: 5=1x5 and that's it. 13=1x13 and that's it.
But 6=1x6 and 2x3. 18=1x18, 2x9, and 3x6. So both 6 and 18 are composite and not prime.
So here we've discover a real (and important) difference between these two groups of numbers.
I thought it was interesting that in the movie 'Contact', the extraterrestrials chose to transmit the sequence of primes as their first contact message with another intelligent species.
Number Theory studies such properties in depth.
There are any number of further statements one can make along these lines.
For example, it is very easy to see that the sequence of prime numbers is an infinite list... and so is the sequence of composites....
So interestingly we've decomposed the infinite list of whole numbers into two different infinite sublists.
We can go even further. For example, one can show that there are infinitely many prime numbers that leave a remainder of 1 when divided by 4: like 5, 13, 17, 29, 37, 41, etc.
But there are also infinitely many primes that leave a remainderof 3 when divided by 4: 3, 7, 11, 19, 23, 31, 43, etc.
Notice that these lists are different and that aside from 2, these two infinite lists exhaust the infinite list of primes. Every prime will appear in one and exactly one of these lists.
Furthermore, you can differentiate between two numbers based solely on their divisibilty properties.
You can play these games forever. We've only scratched the surface in terms of looking at ways to differentiate between numbers.
Far from being trivial, many throughout the ages have asserted that God is, in fact, the ultimate mathematician.
I'll come back to this topic.