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Numbering should start at zero (1982)

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Re: Numbering should start at zero (1982)

#301

Post by nivertech »

jmkr wrote: post #299
nivertech wrote: post #182

Earlier quoted context omitted.

IMO zero represents an absence of quantity and doesn't appear in Nature, so it cannot be classified as a Natural number Just like a negative numbers, it's a higher-level abstraction or a model, not a direct observation from the Nature Likewise, the digit "0" originating from the Hindu-Arabic numeral system[1] is merely a notation, not a number --- 1. https://en.wikipedia.org/wiki/Hindu%E2%80%93Arabic_numeral_s...

A symbol being arbitrary doesn't influence the reality of the meaning behind a thing. I've always thought about `zero` while counting, it never was about `0`. I observe zero. I don't think zero is an absence of quantity. I don't think zero is the null set. You can write types in a programming language, but there are other type theory books that do include zero in the natural numbers. And type theory comes from number…

> I don't think zero is the null set.

zero is the cardinality of the empty set

> I observe zero.

it cannot be observed directly at any static point in time, but it can be observed as a dynamic process when some quantity goes down to empty and back up over time

> In fact I'd be happy to write `>=0` or `>0` or `=0` any day instead of mangling the idea of zero representing 0 and zero representing something like `None`, `null` or any other tag of that sort. I don't think the natural world has anything like "nothing" it just has logical fallacies.

N, W, R, etc. - r just well-known names for sets of numbers, nothing stops us from defining better or additional names for them (with self-describing names)

We can discuss Empty type[1] vs Unit type[2], but I think it goes off-topic

---

1. https://en.wikipedia.org/wiki/Empty_type

2. https://en.wikipedia.org/wiki/Unit_type

Re: Numbering should start at zero (1982)

#302

Post by JadeNB »

jacksnipe wrote: post #294
JadeNB wrote: post #286

Earlier quoted context omitted.

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…

It’s not really considered throwing away Peano’s work. Peano was working in the very infancy of logical formalism. As it turns out, further work developing on his discovered that using a recursively enumerable schema for induction rather than a second-order induction axiom gives rise to a simpler abstraction that still has all the properties that Peano actually desired, and which makes further developments in the spa…

> Continuing to call it Peano Arithmetic is respect for the fact that the guy got it mostly right, and it took the mathematics community many more years to refine the ideas to their current point.

> Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so would Peano.

Oh, by no means do I object to calling an updated and generalised version by the name of the person who originated the subject! Since you've brought up Galois, I hardly think that he'd recognize the modern theory of Galois connections, but I think that the name is wholly appropriate.

No, what I thought was a shame is if the original theory doesn't get discussed at all. If my only exposure to Peano's work was, for example, the axiom schema in Enderton, then I don't think I'd be able to appreciate why it's such a big deal. That would feel to me like teaching the theory of Galois connections without ever saying anything about field theory! Whereas, on the other hand, I did immediately understand as an undergraduate the magic of being able to define everything in terms of the pointed set using induction, and I think I'd appreciate even more having seen that first and then seeing how it is updated for modern mathematical logic.

In fact, at a casual glance, I still don't see why L1, L3, and the A, M, and E axioms can't be omitted in the presence of the axiom(s) on p. 269, which has been the whole substance of my objection. I believe that there's an answer, but, if I don't see it as a professional mathematician (though not a logician), then surely it can't be true that every undergraduate will appreciate it!

Re: Numbering should start at zero (1982)

#303

Post by School-Cotton »

zajio1am wrote: post #283
ziddoap wrote: post #217

Earlier quoted context omitted.

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.

I'm a native English speaker and whether it's harder to read depends on what you mean by "harder". It's immediately obvious what it means (so it's not "harder" in that sense), but on the other hand, it's jarring and distracts me from the point being made.

Re: Numbering should start at zero (1982)

#304

Post by SideQuark »

nivertech wrote: post #300
SideQuark wrote: post #298

Earlier quoted context omitted.

> using this definition everything is Natural, including fore example Complex numbers, which is obviously incorrect, and thus invalidates yr argument Except you’re wrong here; should we thus call your argument “obviously incorrect”? Complex numbers are natural; they’re fundamental in quantum mechanics. Ever since Schrödinger’s equation fundamentally required them for time evolution of states, physicists (and philosop…

No doubt they're useful to explain some theories, but did someone observed in the Nature 6-4i apples or -7+3i particles?

They’re more than useful: they’re required, hence the research demonstrating it that I linked. That you don’t understand it doesn’t mean the rest of science is at your level of ignorance on the topic.

Your repeated, willful ignorance on a topic, especially when shown to you, is why you have such low understanding of the incorrect claims you make.

Take a moment and learn. Then maybe you’ll not repeat claims shown to be wrong.

Re: Numbering should start at zero (1982)

#305

Post by nivertech »

SideQuark wrote: post #304
nivertech wrote: post #300

Earlier quoted context omitted.

No doubt they're useful to explain some theories, but did someone observed in the Nature 6-4i apples or -7+3i particles?

They’re more than useful: they’re required, hence the research demonstrating it that I linked. That you don’t understand it doesn’t mean the rest of science is at your level of ignorance on the topic. Your repeated, willful ignorance on a topic, especially when shown to you, is why you have such low understanding of the incorrect claims you make. Take a moment and learn. Then maybe you’ll not repeat claims shown to b…

I never agreed or disputed whether they're required or not, what I asked was whether they (complex numbers) were ever observed in Nature

Re: Numbering should start at zero (1982)

#306

Post by jacksnipe »

JadeNB wrote: post #302
jacksnipe wrote: post #294

Earlier quoted context omitted.

It’s not really considered throwing away Peano’s work. Peano was working in the very infancy of logical formalism. As it turns out, further work developing on his discovered that using a recursively enumerable schema for induction rather than a second-order induction axiom gives rise to a simpler abstraction that still has all the properties that Peano actually desired, and which makes further developments in the spa…

> Continuing to call it Peano Arithmetic is respect for the fact that the guy got it mostly right, and it took the mathematics community many more years to refine the ideas to their current point. > Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so w…

Oh, then yeah, I totally agree. In general I think it's a shame that so little emphasis is placed on the history of mathematics, though at the same time I appreciate that most of my peers just didn't care :(

Re: Numbering should start at zero (1982)

#307

Post by jacksnipe »

JadeNB wrote: post #302
jacksnipe wrote: post #294

Earlier quoted context omitted.

It’s not really considered throwing away Peano’s work. Peano was working in the very infancy of logical formalism. As it turns out, further work developing on his discovered that using a recursively enumerable schema for induction rather than a second-order induction axiom gives rise to a simpler abstraction that still has all the properties that Peano actually desired, and which makes further developments in the spa…

> Continuing to call it Peano Arithmetic is respect for the fact that the guy got it mostly right, and it took the mathematics community many more years to refine the ideas to their current point. > Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so w…

Second addendum, chapter 4 is about second order logic and apparently I just forgot that exercise 1 is simply showing that you get all of the structure built up in Chapter 3 with Peano's original formulation in second-order logic. Seems that here I'm the one suffering from a lack of historical context!

I think from a logic standpoint this also makes sense -- getting to undecidability quickly makes taking the direct route through first-order logic more appealing.

If I'm being honest, I now do feel a little bit deprived, I probably would have enjoyed the categorical view when I was learning this too.

Re: Numbering should start at zero (1982)

#308

Post by SideQuark »

nivertech wrote: post #305
SideQuark wrote: post #304

Earlier quoted context omitted.

They’re more than useful: they’re required, hence the research demonstrating it that I linked. That you don’t understand it doesn’t mean the rest of science is at your level of ignorance on the topic. Your repeated, willful ignorance on a topic, especially when shown to you, is why you have such low understanding of the incorrect claims you make. Take a moment and learn. Then maybe you’ll not repeat claims shown to b…

I never agreed or disputed whether they're required or not, what I asked was whether they (complex numbers) were ever observed in Nature

Yes, they have been observed in the same rigor as any number you claim has been observed. If you’d spend a moment and read instead of repeated willful ignorance, you’d learn something.

Now go do your homework. Attaching idiot phrases like complex apples is as stupid as claiming we don’t see radio waves so they can’t exist or that matter cannot be mostly empty space because you can stack books.

Your limited imagination, understanding, and unwillingness to learn, even when given a source and phrases to look into, doesn’t apply to those scientists that have done the work.

Re: Numbering should start at zero (1982)

#309

Post by nivertech »

SideQuark wrote: post #308
nivertech wrote: post #305

Earlier quoted context omitted.

I never agreed or disputed whether they're required or not, what I asked was whether they (complex numbers) were ever observed in Nature

Yes, they have been observed in the same rigor as any number you claim has been observed. If you’d spend a moment and read instead of repeated willful ignorance, you’d learn something. Now go do your homework. Attaching idiot phrases like complex apples is as stupid as claiming we don’t see radio waves so they can’t exist or that matter cannot be mostly empty space because you can stack books. Your limited imaginatio…

> Yes, they have been observed

Any references?

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