## The parity value of 0

Use this forum to discuss the philosophy of science. Philosophy of science deals with the assumptions, foundations, and implications of science.
Alan Masterman
Posts: 109
Joined: March 27th, 2011, 8:03 am

### The parity value of 0

Why do some mathematicians feel a pathological need to prove that 0 is an even number? Nothing of real importance in mathematics depends upon it. The very few contexts which treat 0 as even could be equally well served by a contextual definition, without the need to prove evenness in any absolute or universal sense.

There is a popular website which summarises in one article all of the "proofs" that 0 is even. They are generally specious, and should not be cited; some are so brazenly fallacious, it's difficult to believe they are offered in good faith. The website in question - you may guess which one - is NOT generally regarded as a reference of academic standard.

A proof that 0 has NO parity value, on the other hand, would serve at least one useful purpose: it would dispel the dense intellectual fog which surrounds this question!

If you feel disposed to dignify my own intellectual fogginess with a response, please first reflect:

(1) Parity value is a property of the natural numbers.
(2) The natural numbers are logically primitive.
(3) Therefore, whatever process determines parity value, operates at the logically primitive level.
(4) (HINT! Bone up on the axioms...)
JackDaydream
Posts: 1802
Joined: July 25th, 2021, 5:16 pm

### Re: The parity value of 0

@Alan Masterman

I don't consider myself as a mathematician, so I am approaching your question from a purely reflective angle. 0 is about nothing. So, it a basic position of absence, so to see it as even would be about seeing nothing as even. Within binary logic, ita is about the negatives and positives, like the yin and the yang.

I hope that you get some mathematical people engaged in your thread. I just gave you my basic thoughts, with a view to getting the thread going and hoping that other people will contribute...
GE Morton
Posts: 3664
Joined: February 1st, 2017, 1:06 am

### Re: The parity value of 0

Alan Masterman wrote: January 13th, 2022, 12:14 pm Why do some mathematicians feel a pathological need to prove that 0 is an even number? Nothing of real importance in mathematics depends upon it. The very few contexts which treat 0 as even could be equally well served by a contextual definition, without the need to prove evenness in any absolute or universal sense.

There is a popular website which summarises in one article all of the "proofs" that 0 is even. They are generally specious, and should not be cited; some are so brazenly fallacious, it's difficult to believe they are offered in good faith. The website in question - you may guess which one - is NOT generally regarded as a reference of academic standard.

A proof that 0 has NO parity value, on the other hand, would serve at least one useful purpose: it would dispel the dense intellectual fog which surrounds this question!
No proof that 0 is even is necessary; it is even by definition: an even number is an integer which may be divided by 2 with no remainder. Since 0 satisfies that definition it is even.

Nor is any proof possible that 0 has no parity. Parity is the property of being even or odd. Since 0 is even, it has a parity value (again) by definition.
Alan Masterman
Posts: 109
Joined: March 27th, 2011, 8:03 am

### Re: The parity value of 0

"No proof that 0 is even is necessary; it is even by definition: an even number is an integer which may be divided by 2 with no remainder. Since 0 satisfies that definition it is even."

"Nor is any proof possible that 0 has no parity. Parity is the property of being even or odd. Since 0 is even, it has a parity value (again) by definition"

GE, in mathematics, a definition is not something you "prove"; it is merely a statement to clarify the meaning of a technical term. You then go on in the same sentence to supply a highly questionable proof of something which you have just asserted is a definition.

Remember that we are dealing with a primitive property which, I have argued, must be discussed at the axiomatic or "primitive" level. At this level, the function 0/2=0 is unintelligible; there is no number in the natural number line which is equal to half of 0, and the axioms forbid the same number to be equal to both itself and half of itself.
Alan Masterman
Posts: 109
Joined: March 27th, 2011, 8:03 am

### Re: The parity value of 0

Jack, I acknowledge your reply, by saying that I think you have accurately perceived the philosophical heart of the problem in recognising that 0 represents "nothing". I don't think that everybody in the mathematical forum understands exactly what "nothing" means. But it occurs to me now that my original post was not properly thought through; I think I should restart the thread from a more fundamental level.
GE Morton
Posts: 3664
Joined: February 1st, 2017, 1:06 am

### Re: The parity value of 0

Alan Masterman wrote: January 16th, 2022, 6:55 am
GE, in mathematics, a definition is not something you "prove"; it is merely a statement to clarify the meaning of a technical term. You then go on in the same sentence to supply a highly questionable proof of something which you have just asserted is a definition.
You just contradicted yourself. If a definition is "not something you prove" (which it is not), then there is no need to "supply a highly questionable proof of something which you have just asserted is a definition."
Remember that we are dealing with a primitive property which, I have argued, must be discussed at the axiomatic or "primitive" level.
Definitions are the most primitive level. They are stipulated and assumed a priori.
At this level, the function 0/2=0 is unintelligible; there is no number in the natural number line which is equal to half of 0 . . .
Yes, there is, namely, 0.
and the axioms forbid the same number to be equal to both itself and half of itself.
Only for numbers >0. By definition.
ernestm
Posts: 138
Joined: March 5th, 2018, 4:27 am

### Re: The parity value of 0

I'd have to say zero is in interesting quantity.

Mathematics conventionally regards zero as not having a sign. In fact in computers, due to the nature of their binary computation, then can reach a mathematical result of -0. Thus if one holds maths is a priori true, then one also has to admit that there is an alternate form of mathematics, 'practical' rather than 'dieal,' which has different rules.
Alan Masterman
Posts: 109
Joined: March 27th, 2011, 8:03 am

### Re: The parity value of 0

Two comments. Firstly, I respectfully ask respondents to reply to my later post rather than this one, which I now see is defective in a number of ways.

Secondly, I would remind respondents that counter-assertion - particularly, assertion of received or conventional wisdom - is not argument. As Hegel said, "you cannot defeat the enemy where he is not". If you are going to reply, please be ready with proofs and reasoned arguments.
Alan Masterman
Posts: 109
Joined: March 27th, 2011, 8:03 am

### Re: The parity value of 0

I agree that 0 is a fascinating number! And I can easily understand why mathematicians, up until the end of the 19th C, were reluctant to include it among the set of the natural numbers. It has so many interesting properties which are unique to itself. it is the only number which is:

1. the number of a singular or unique set;
2. neither negative nor positive;
3. neither odd nor even ;
4. less than every natural number which is not itself;
5. neither divisible nor multipliable ;
6. common to every type of number line.

Not all of these properties hold true in general or post-axiomatic arithmetic, of course. But then, arithmetic allows us to get away with a lot of exploits which number theory would frown upon.
Alan Masterman
Posts: 109
Joined: March 27th, 2011, 8:03 am

### Re: The parity value of 0

Reasons to be happy that 0 is a number, Part 2:

(1) 0 has a definable magnitude which allows us to position it in the number line.

(2) 0 is computable, meaning that we can incorporate it into any ordinary arithmetical transaction (with certain exceptions which do not affect its status as a number) without generating any contradiction or non-sequitur; for example, 2 x 11 = 22 and 2 x 10 = 20 are logically identical operations of multiplication.

(3) without 0, we would need to invent a special symbol for every number which is a multiple of 10; and we would need a dictionary as large as the universe to record them all. Positional notation would be terrifically cumbersome without a 0 symbol, and functionally impossible for any really large numbers.

(3) 0 enables a proof that the natural number line is infinite. How many numbers are there up to and including the number 9? If 0 is included, the answer must be 10, since 0 must be included among the total. We may generalise this to the the rule that, for any value n, the total quantity of numbers up to and including n will be equal to n+1. And if we define n as "the largest imaginable number"...?
Raymond
Posts: 317
Joined: January 23rd, 2022, 6:47 pm

### Re: The parity value of 0

Zero is the most even number you can('t) imagine. It lays in the exact middle of all positive evens and negative evens. Two away from 2 and -2, Four away from 4 and -4. Two is less even. It lays 2 away from 4 and 6 away from -4.

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