Philosophy Now Forum

I would contend yes.

For example we have the number one. Now I ask you to put all the other numbers out of your mind and focus on the number one. Since we don't have the other numbers to compare with, then quantity has no meaning with the number one. The same with the number two when we don't have other numbers in our universe to compare with, etc.

Let's look at the cubic numbers such as 0, 1, 8, 27... The average person would see cubic numbers. However if you skip the even-positioned cubic, you get 1, 27, 125, 343... and when you sum them, you get 28, 153, 496... which, by algebra, is the triangular number series. If you look at the cubic numbers again and sum them, you get
1, 9, 36, 100... which is the square of the sum of the natural numbers which implies a connection between the sum of the square of the numbers and the triangular number series.

Another way of extending the concept of number is complex numbers. What the extent for the concept of number beyond the examples I've given, I can only guess.

What do you think?

PhilX

Philosophy Explorer wrote: ↑Mon Nov 13, 2017 10:49 pm I would contend yes.

For example we have the number one. Now I ask you to put all the other numbers out of your mind and focus on the number one. Since we don't have the other numbers to compare with, then quantity has no meaning with the number one. The same with the number two when we don't have other numbers in our universe to compare with, etc.

Here it seems that you are saying that without relation to other numbers, an individual number has no meaning. Let us assume that it is true for the sake of argument. But if I were to ask, if we abstract from the relation to other numbers and focus on that-which-results (after the abstraction), can "it" still be said to be a number, given that it has thus no meaning? For we can say that a number has a meaning.
If you say "yes", then you would be saying that a number has no meaning! And if you say "no", then you would not be able to say that this example proves that number groups have more meaning than individual numbers; as that-which-results and which has no meaning is not a number!

You may disagree with me on the following, but I find it quite intuitive to think that the properties of a group of numbers are also those of any of the numbers constituting the group. Moreover, some members of a group may belong to other groups as well, and have properties which other members of any given group do not have. Hence, this would imply that individual members of a group may have more properties than a given group as a whole.

__________

PhilX wrote:What do you think?

I think the examples you gave are interesting. I made some research and found a couple of proofs for these. If you agree, allow me to share these.

Concerning triangular numbers an important result is that the sum of the first n natural numbers give the nth triangular number.

(i=1, i=n)∑ i = n.(n+1)/2

A triangular number should always be able to be expressed in the form n.(n+1)/2 or n.(n-1)/2

PhilX wrote:Let's look at the cubic numbers such as 0, 1, 8, 27... The average person would see cubic numbers. However if you skip the even-positioned cubic, you get 1, 27, 125, 343... and when you sum them, you get 28, 153, 496... which, by algebra, is the triangular number series.

Yes.
Proof:

Sequence: 1^3, 3^3, 5^3...,(2n-1)^3

(i=1, i=n)∑ (2n-1)^3 = n^2.(2.n^2 - 1)-------(1)

A triangular number should always be able to be expressed in the form n.(n+1)/2 or n.(n-1)/2

Rearranging the result of formula (1), we get [2.n^2 * (2.n^2 -1)] /2, which has the form N.(N-1)/2 where N=2.n^2. Hence the sum of the first cubes of odd numbers is a triangular number.

Now, I have computed some of the values for n in the above formula, and found something interesting.

Some results:
For n=1, 1 * (2-1) = 1*1 =1. The first triangular number is one; T1 =1

For n=2, 2^2 * (2(2^2) -1) = 4 * 7= 28. The seventh triangular number is 28; T7= 28

For n=3, 3^3 * (2(3^2)-1) = 9 * 17 =153. T17 = 153.

For n=4, 16 * 31 = 496. T31= 496. etc...

The factor (2.n^2 - 1) gives you the position of the resulting triangular number in the sequence of triangular numbers. And also each time we are multiplying by a square. That's it! It might not be much, but I found it to be interesting!

PhilX wrote:If you look at the cubic numbers again and sum them, you get
1, 9, 36, 100... which is the square of the sum of the natural numbers which implies a connection between the sum of the square of the numbers and the triangular number series.

1^3 +2^3 + 3^3 +....n^3 = (1+2+3+...+n)^2

I find this to be a beautiful result.

Wikipedia presents one of the proofs here.