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Numbering should start at zero (1982)

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Re: Numbering should start at zero (1982)

#281

Post by InsideOutSanta »

adrian_b wrote: post #224
InsideOutSanta wrote: post #198

Earlier quoted context omitted.

> Your arguments using "first" and "second" are invalid, > because those words have nothing to do with numbers "First" is the ordinal number corresponding to the number one, while "second" is the ordinal number corresponding to the number two. You can represent "first" as 1st and "second" as 2nd. I believe the origins of these two words do not significantly impact my argument.

Nothing in the word "first" indicates that it corresponds to "one". The same for "second". Those words are not numerals, "first" is a superlative adjective, while "second" is an active participle. They are perceived as ordinal numerals only because English does not have ordinal numerals for the 2 initial positions of a sequence and "first" and "second" are used instead of the missing ordinal numerals. The abbreviatio…

It doesn't matter whether there is a long tradition of correspondence between first and 1 and second and 2. What matters is that the correspondence exists today, because I'm making my argument today.

Re: Numbering should start at zero (1982)

#282

Post by pfortuny »

jonahx wrote: post #250
pfortuny wrote: post #231

Earlier quoted context omitted.

Actually, the duality arises from counting (there can be 0 items) and ordering (there is only a 1st item), conceptually. Which is why the year 2000 can and cannot be the start of the 3rd millenium, for instance.

I don't follow the distinction you're making. The number line is ordered and contains a 0.... The GP's explanation seems more fitting for the year 2000 ambiguity. Are you measuring completed years (celebrate millenium on NYE 2001) or are years the things happening between 0 and 1, 1 and 2, etc (celebrate on 2000, because we're already in the 2000th "gap")?

Order as in "first, second, third", whereas counting is "none, one, two".

This is not a formal distinction, it is a conceptual one for things (not mathematical models).

Re: Numbering should start at zero (1982)

#283

Post by zajio1am »

ziddoap wrote: post #217
Brian_K_White wrote: post #202

Earlier quoted context omitted.

I want to be rude. The mix of "The same consideration applies to coordinate systems:" and "r u positioning urself..." in the same message is ridiculous to me and I can't take anyone seriously who speaks like this. It's not illegal or immoral, but it is at the very least, demonstrably distracting from their own actual substantive point. Here we all are talking about that when they otherwise had a perfectly good observ…

People taking such offense to something so absolutely inconsequential, on an internet forum no less, is ridiculous to me and I can't take anyone seriously who gets worked up about it. You, and parent poster, understood them fine. You, and the parent poster, are the ones who are steering the conversation in the direction of how they typed, not what they typed. They had a "perfectly good observation to talk about", yet…

I do not know about native speakers, but for me as a non-native speaker, such informal language is much harder to read.

Re: Numbering should start at zero (1982)

#284

Post by zajio1am »

Aransentin wrote: post #46

Perhaps ideally we'd change English to count the "first" entry in a sequence as the "zeroth" item, but the path dependency and the effort required to do that is rather large to say the least. At least we're not stuck with the Roman "inclusive counting" system that included one extra number in ranges* so that e.g. weeks have "8" days and Sunday is two days before Monday since Monday is itself included in the count. *…

Note that 'first' and 'second' are not etymologically related to one or two, but to 'foremost'. Therefore, it is would make sense to use this sequence of ordinals:

first, second, twoth, third, fourth, ...

or shortened:

0st, 1nd, 2th, 3th, 4th ...

Re: Numbering should start at zero (1982)

#285

Post by zajio1am »

lupire wrote: post #261
myfonj wrote: post #11

I found it devastating that there are no distinct agreed-upon words denoting zero- and one-based addressing. Initially I thought that the word "index" clearly denotes zero-base, and for one-base there is "order", "position", "rank" or some other word, but after rather painful and humiliating research I stood corrected. ("Index" is really used in both meanings, and without prior knowledge of the context, there is real…

"offset" and "ordinal".

Ordinals in math also start at zero. It is just common people accepted zero for cardinals and but not (yet) for ordinals.

Re: Numbering should start at zero (1982)

#286

Post by JadeNB »

jacksnipe wrote: post #264
JadeNB wrote: post #263

Earlier quoted context omitted.

In the usual terminology, these are not axioms; as your wording itself says, they are definitions. (Indeed, I'd argue that it's almost ungrammatical to say something is "defined in the axioms"; axioms may, and probably must, be stated in terms of definitions, but the definitions are not themselves axioms.) As I say, one can quibble about terminology, since what's important is less what's axiom and what's definition,…

No, they are axioms. Peano arithmetic itself is a first-order theory, and a theory is just a recursively enumerable set of axioms. Enderton, “A Mathematical Introduction to Logic, 2nd Ed.”, p,203,269-270 Kleene, “Mathematical Logic”, p.206 EDIT: It seems like you're talking about Peano's original historical formulation of arithmetic? That's all well and good but it is categorically not what is meant by "Peano Arithme…

I'm away from my library, but fortunately the books you referenced are a Google away, so I could consult them and confirm that they say what you say. I'm not quite willing to accept Kleene's word as an authority on common modern mathematical practice, since he was a theoretical computer scientist before the term, but, though I'm not familiar with Enderton's book, it certainly looks like a reasonably standard one.

But these are all referring to Peano arithmetic as a model of the theory of the natural numbers. And that seems a bit silly: the impact of Peano's work wasn't because he showed that there was a model of the theory of the natural numbers, which everybody believed if they bothered to think about it, but because he showed that all you needed to make such a model was a successor operation satisfying certain axioms. Yes, they may be less model-theoretically congenial because they're second order, but to change Peano's work from what he did historically and still call it Peano's seems strange to me. (I'm fine with dressing it up in modern language, and calling it an initial object in the category of pointed sets with endofunctor, which perhaps is biased but still seems to me to be capturing the essential idea.)

Certainly I was taught the second-order approach, though it was as an undergraduate; I've never taken a model-theory class. As I say, I'm away from my library and so can't consult any other sources to see if they still teach it this way, and anyway I am a representation theorist rather than a logician; but, if the common logical approach these days really is to discard Peano's historical theory and to call by Peano's name something that isn't his work, even if it is more convenient to use, then I think that's a shame from the point of view of appreciating the novelty and ingenuity of his ideas. But just because I think something is a shame doesn't mean it's not true, and so far you've produced evidence for your view and I can't for mine, so I can't argue any further.

Re: Numbering should start at zero (1982)

#287

Post by Brian_K_White »

c22 wrote: post #259
Brian_K_White wrote: post #202

Earlier quoted context omitted.

I want to be rude. The mix of "The same consideration applies to coordinate systems:" and "r u positioning urself..." in the same message is ridiculous to me and I can't take anyone seriously who speaks like this. It's not illegal or immoral, but it is at the very least, demonstrably distracting from their own actual substantive point. Here we all are talking about that when they otherwise had a perfectly good observ…

Stuff like this could be an innocent attempt to obfuscate one's stylometric fingerprints. I just read around it like mistakes from ESL commenters.

agreed and good to remember, thanks

Re: Numbering should start at zero (1982)

#288

Post by JadeNB »

jacksnipe wrote: post #264
JadeNB wrote: post #263

Earlier quoted context omitted.

In the usual terminology, these are not axioms; as your wording itself says, they are definitions. (Indeed, I'd argue that it's almost ungrammatical to say something is "defined in the axioms"; axioms may, and probably must, be stated in terms of definitions, but the definitions are not themselves axioms.) As I say, one can quibble about terminology, since what's important is less what's axiom and what's definition,…

No, they are axioms. Peano arithmetic itself is a first-order theory, and a theory is just a recursively enumerable set of axioms. Enderton, “A Mathematical Introduction to Logic, 2nd Ed.”, p,203,269-270 Kleene, “Mathematical Logic”, p.206 EDIT: It seems like you're talking about Peano's original historical formulation of arithmetic? That's all well and good but it is categorically not what is meant by "Peano Arithme…

> EDIT 2: This is all a digression anyway. Both first- and second-order PA label the start of the Z-chain as 0; so any model of PA contains 0 when interpreted as a model of PA.

Ah, good point that this was the actual source o# the discussion. This one at least can be argued, because the question is about how things should be axiomatized/defined, not how they are. And certainly the theory of the "natural numbers starting with 1" can be axiomatised just as well as the "natural numbers starting with 0." All these axioms are made by humans, and an appeal to existing axioms here can only say what's been done, not what should be. (And I say this as someone who does start my naturals at 0.)

Re: Numbering should start at zero (1982)

#290

Post by [deleted] »

zajio1am wrote: post #284
Aransentin wrote: post #46

Perhaps ideally we'd change English to count the "first" entry in a sequence as the "zeroth" item, but the path dependency and the effort required to do that is rather large to say the least. At least we're not stuck with the Roman "inclusive counting" system that included one extra number in ranges* so that e.g. weeks have "8" days and Sunday is two days before Monday since Monday is itself included in the count. *…

Note that 'first' and 'second' are not etymologically related to one or two, but to 'foremost'. Therefore, it is would make sense to use this sequence of ordinals: first, second, twoth, third, fourth, ... or shortened: 0st, 1nd, 2th, 3th, 4th ...

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