0

Foreword:

0 is considered an even number,
but if 0 would be an even number,
then 0 apples would count an even number of apples.

Example:

 3 apples [๐Ÿ๐Ÿ๐Ÿ]  
 2 apples [๐Ÿ๐Ÿ  ]  
 1 apple  [๐Ÿ    ]  
 0 apples [      ]

Can we see 0 apples counting an even number of apples?

Question:
Is this of some kind of sophism example considering 0 to be even?

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  • 2
    What issue do you see with 0 apples being considered an even number of apples? I don't see the problem. That sounds reasonable enough to me.
    – NotThatGuy
    Sep 28 at 22:38
  • I don't see 0 counting any apples. I guess in your logic then 0 isn't even a number?
    – JonathanZ
    Sep 29 at 4:59
  • Numbers being instruments to count things is a nice feature, but not the core of what a number is. Whether or not "zero apples" is a legitimate amount of apples isn't relevent to 0 being even. An integer X, is even if X = 2*Z for some integer Z. As, 0 = 2*0; 0 is by definition an even number. Note: I can't have ฯ€ many apples, and yet Math would be bereft if we needed to exclude numbers like ฯ€ because we can't count apples with it.
    – Michael Carey
    Sep 29 at 18:09

3 Answers 3

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If you are taking sophism to mean a type of fallacy, then no.

The extension of even numbers from outside the naturals to the non-negative integers is an example of a technical definition in mathematics. Math systems like PA, are examples of formal systems, and formal systems are built on axioms. If one considers the trichotomy property of the integers, where a number is negative, positive, or zero, to maintain closure, one has to deal with the pesky case of zero. (Think of the definition as a z/2 must be in Z.) Obviously, for positives and negatives, to define a double of an integer as zero is fairly intuitive. For any double which is positive or negative, dividing by two results in another positive or negative respectively. But what should be done with zero? It just so happens that because zero occurs before and after an odd number, it just makes sense consider it even, since it is divisible by two. In this way, the formula 2z is closed.

The idea that mathematical systems are built up from axioms is central to mathematical constructivism (SEP), and finding axioms to make systems functional or better is known as reverse mathematics. You may hear terms like abstract objects or fictions thrown around in nominalist (SEP) or fictionalist mathematical (SEP) thinking. This is in distinction to Platonic reasoning about mathematics which tends to see numbers as real or important to identify with real entities. More modern thinking defines numbers not as names to corresponding to counts, but more abstractly like iterations of the successor function.

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Start with any even number. Adding/subtracting 2 should take you to the next higher/lower even number.
8 - 2 = 6, 6 - 2 = 4, 4 - 2 = 2, 2 - 2 = 0. O is an even number.

Doing the same for odd numbers you get ...
7 - 2 = 5, 5 - 2 = 3, 3 - 2 = 1, 1 - 2 = -1. 0 is not an odd number.

EDIT START

If one is still unsure, we can always specify the domain like so:
a) 0 is the first even whole number and 1 is the first odd whole number.
b) 1 is the first odd natural number and 2 is the first even natural number

The issue, it seems, is our starting point (0 or 1).

EDIT END

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Definition:
Any integer counts a number of units 1.
The number of units is odd, if divisible by 2 with remainder.
The number of units is even, if divisible by 2 without remainder.
Zero counts zero units 1, therefore 0 has neutral parity, neither odd nor even.

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  • This ia false. Zero is an even number. An Integer x is even if it can be written as a product of 2 and another integer y. As 0 can be written as 2*0; 0 is even. What you might be thinking of is signs. 0 is neither positive nor negative
    – Michael Carey
    Sep 29 at 18:06
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    Sep 30 at 9:35

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