In multiplying one (1) is neutral element: a * 1 = a. For example, 7 * 1 = 7. Number one keeps the identity of a number, which includes a number being even or uneven. But what about zero (0)?
0 * 1 = 0. Does one keep the identity of zero or does zero keep the identity of its own? The property of this identity is ”zeroing” property: a * 0 = 0, were a whatever real number, including one and on the other hand zero.
In case -2 * 0, zero takes the whole identity of number -2: The number being negative and even; as a result we get ”just” zero. Similar happens in 2 * 0 = 0.
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My two cents: Zero ”zeroes” any number except itself. It ”zeroes” the whole identity – including a number being even or uneven – of any number except from itself; in case 0 * 0 = 0 zero keeps the identity of its own, it doesn’t ”zero” itself, which reflects the identity of zero itself, how it is neutral in a deep sense and meaning.
Mathematical philosophically zero refers to none, there isn’t something at all. Still, zero refers different than nothing. As I’ve written before emptyness (”zero”) can be created, nothing can’t be created; it is from which the creation begins.
Let us assume, that we have two (2) coins. It’s even amount of coins. Let’s give one coin to a poor beggar. Now we have only one coin, uneven amount of coins. We’ll give that coin to a poor beggar too. Now we have no coins at all, the number of coins we have is zero. Do we have still again even number of coins, as we have zero amount of coins? I mean, we don’t have coins left at all!
The coins we had were in a wallet and the two coins were all we had there; now the wallet is empty. Is emptyness even or uneven? Or are we speaking now about different matter?
As far as I can see, if the number of something is different than zero, there must exist something, somehow. This number is even or uneven.
So, number being even or uneven, philosophically would refer to existence; something must somehow exist, that is, the number of something is different than zero. This amount can be negative or positive, even or uneven, but not zero.
But if something doesn’t exist, the amount of this something is zero, that isn’t even or uneven, as stated before. If the “number of something” is even or uneven, something must exist, somehow.
Technically one test to determine, that is a number even, is to divide the number to be tested by 2; if reminder is zero, the number is even. This test is suspicious to zero from two (2) reasons:
- 0 / a = 0 anyway were the number a whatever real number (except zero)
- Two (2) is greater than zero by its absolute value (philosophical mathematical problem)
My two cents: Zero is neutral element in addition and one of its properties is, particularly philosophically, that as to being even or uneven, it is neutral.
(As a sidenote something came into my mind from section 2 above: Is number one (1) somehow fundamentally uneven in natural numbers set?)