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AN AXIOMATIC TEST OF PARITY VALUE

The primitive natural number line is elaborated in parallel with the axioms of arithmetic. It is defined as the series 0 – 9; 0 is further defined as "not the successor of any number", 1 as the successor of 0, 2 the successor of 1, and so on. We symbolise this relation in the function S(), which evaluates to the number next after the number given in the argument; for example, S(0)=1. So, the logical structure of the natural number line is:

0, S(0), S(S(0)), S(S(S(0))), S(S(S(S(0))))…

The interval between each number and the next is self-evidently a unity; so, it must be true that:

S(S(0)) = S(0) + S(0)

This is important because it reveals the process by which the parity values of numbers become fixed at the axiomatic level. Those like S(S(0)), which we can analyse into two similar instances of a smaller cardinality, we choose to call "even". Those which are analysable only into dissimilar instances we call "odd".

It is obvious that this process cannot assign a parity value for 0, because 0 is not the successor of any number. There are now two options.

(1) We might try to devise a new test, or a modification of this test, which can encompass all of the numbers including 0 (without straying outside the context of the axioms, because parity value is determined, as we have just seen, at the axiomatic level). The main difficulty will be that "parity" is defined as a similarity of two subsets. The number 2 = 1 + 1; the original cardinality is distributed evenly between two subsets, which thereby attain to parity, so we say 2 is "even". The number 3, on the other hand, can approach no closer to parity than 2 + 1; the subsets are an odd pair, so we say 3 is "odd". No subsets of either kind can be identified for 0, because there is no smaller number in the number line. The only way forward, perhaps, is to shift the logical goalposts and allow 0 = 0 + 0; but we see at once that 0 may be equal to ANY number of 0's – there is nothing unique or privileged about 0 = 0 + 0; so it is difficult to see how this might lead us to an intelligible proof of parity value. (It would also imply a fundamental rewrite of set theory, which does not permit division of the null set).
(2) The other option is to accept the argument from succession at face value as the sole criterion of parity value, because it is self-evidently simple, valid, and presents no logical difficulties, at the modest price of conceding that 0 is neither odd nor even.

The parity value of 0 - if it has one - is a surprisingly difficult philosophical question. To address it properly, without even guaranteeing a successful resolution, presupposes a reasonable familiarity with basic set theory and with the Dedekind/Peano axioms, which I respectfully ask respondents to bear in mind. Please do not reply with arguments lifted from THAT website.

In conclusion, may I say the reason why 0/2=0 "works" for post-axiomatic arithmetic is philosophically interesting in its own right, but does not constitute a proof of parity value.

First, you know that we're simply talking about stuff we're making up, right?

Mathematics is a very abstract way to think about relations. We invent a language to represent the abstract (largely quantificational) way that we're thinking about relations.

Well, that landed like a bomb! Is there NOBODY in this forum who feels qualified to comment on the Axiom of Induction or basic set theory?

Let me make a fresh start. Why is 0/2=0 NOT a proof of evenness? Because it is an iron rule of arithmetic that for every non-zero value of n, 0/n=0. It is important to fix in mind that the proof of this theorem does NOT in any way hinge upon parity value. There is no logical connection whatsoever. So it doesn't matter a rat's patootie whether 0 is even, or odd, or has no parity value; 0/2=0 will be true in any case. Consequently, 0/2=0 can not prove evenness.

We may approach the same conclusion from another direction. Let it be given that 0 is provably odd: will 0/2=0 be rendered false thereby? To answer 'yes', is to commit oneself to argue that (for every non-zero value of n) 0/n=0 is false. To answer 'no' is to concede that 0/2=0 is equally consistent with oddness.

Let me append a comment upon which I would really appreciate some genuine philosophical insight: given that "zero on zero" operations are forbidden by both the Dedekind/Peano axioms and by set theory, why does 0/2=0 appear to work for post-axiomatic arithmetic? Not only does it "work", it is useful and perhaps even necessary, for everyday purposes. My own (tentative) hypothesis is that the quotient is not really 0 but Ø;

0/2 = Ø

- in other words, a null value indicating that the operation of division has failed. Since arithmetic has no criterion to distinguish between Ø and 0, we may happily continue as if the outcome were really 0.

Alan Masterman wrote: ↑February 5th, 2022, 6:06 am Let me append a comment upon which I would really appreciate some genuine philosophical insight: given that "zero on zero" operations are forbidden by both the Dedekind/Peano axioms and by set theory, why does 0/2=0 appear to work for post-axiomatic arithmetic? Not only does it "work", it is useful and perhaps even necessary, for everyday purposes. My own (tentative) hypothesis is that the quotient is not really 0 but Ø;

0/2 = Ø

- in other words, a null value indicating that the operation of division has failed. Since arithmetic has no criterion to distinguish between Ø and 0, we may happily continue as if the outcome were really 0.

As I mentioned before, there is a problem with how central processor units work in computers, that they only permit calculation of numeric quantities. There is no way to represent null values, not any imaginary numbers, nor the sign of zero, because of they way they are built. So there is 'practical alegbra' as computers think, which assesses 0/2 as 0 to avoid error conditions, and there is 'pure algebra' which can make a distinction but which cannot be implemented in real-world machines. It exists only conceptually.

Good point; but is there more which could be said, ernestm, from the philosophical point of view? I say this because we've surely been using 0/2=0 successfully since long before the age of computers. I don't pretend for a moment to see my way through to the end of this question, of course.

Here is an arithmetical statement of the argument I began with:

- For any natural number n: subtract one unit at a time from n, so that n tends towards 0; and, up to any given step, let m be equal to the total quantity thus far subtracted.

(1) If, at some point during this procedure, it occurs that n = m, then n was initially an even number. If n attains to 0 and this condition has not occurred, then n was initially odd.
(2) If n is such that no unit can be subtracted from it (ie n = 0), then there can be no “total quantity subtracted”, and no "m" value against which we can assess parity; and so n will not attain to either odd or even.

And here is the same argument in the vocabulary of basic set theory:

- For any natural number n: from a set A of cardinality n, remove one element at a time to a separate set B.

(1) If, at some point during this procedure, it occurs that B and A can be paired in one-one relation [ie the sets are "similar"/have the same cardinality/attain to "parity"], then n is an even number.
(2) If set A is exhausted and this condition has not occurred, n is odd.
(3) If the cardinality of A is such that no element can be removed from it [ie A is the empty set], then a second set can not be created, the question of parity [of subsets] will not arise, and n will not attain to either odd or even.

It is obvious, I hope, that the argument expresses itself in purely axiomatic terms: "natural number", addition and subtraction, and the relation of similarity (=). Assuming validity, the argument has many important consequences and implications. Here are two on which I especially invite comment:

(a) The operation of division, the number 2, and the concept of “remainder” are not logically necessary to a proof of parity value. Consequently a post-axiomatic or mathematical test which relies on any or all of these - for example the "test of even divisibility by 2", with or without a remainder - can be ONLY a test and not a proof, since the required relation of logical necessity does not exist.
(b) There is a conventional belief that the evenness of 0 can be proved by a test of even divisibility (of which there are many forms). The test I have outlined above helps to blow away some of the intellectual fog surrounding this question by supporting the position that 0 is neither odd nor even.

Critique please, somebody? All comments welcome. Help me out here!