[HN Gopher] The Invention of Zero
___________________________________________________________________
The Invention of Zero
Author : Anon84
Score : 119 points
Date : 2024-07-06 15:32 UTC (3 days ago)
(HTM) web link (www.themarginalian.org)
(TXT) w3m dump (www.themarginalian.org)
| briankelly wrote:
| Interesting enough, the oldest recorded use of the symbol zero in
| mesoamerica is older. From
| https://baas.aas.org/pub/2021n1i336p03/release/2 - "The oldest
| representation of the Mesoamerican zero, dating from the year 31
| BCE, is found in Stela C in the ancestral Olmec site of Tres
| Zapotes in Veracruz, Mexico."
| adrian_b wrote:
| Older than the use of zero in India, but hundreds of years
| later than the use of zero in Mesopotamia.
|
| In any case, this must have been an independent invention of
| zero.
| kragen wrote:
| unless it was brought over by the flying saucers
|
| (now where did i put my aluminum foil)
| adrian_b wrote:
| Speaking about the "invention of zero" is not a good choice of
| words.
|
| All languages have had words for "zero", for at least a few
| thousand years.
|
| For instance English has inherited "null" from Latin, whose
| literal meaning was initially "not even a small one", but whose
| meaning has become "zero". Latin also used very frequently the
| word "non-null", with the meaning "one or more". Latin had a few
| more other words that could be used to express the quantity
| "zero". The same was true for Ancient Greek and for the languages
| for which even older records exist.
|
| What the parent article intends to discuss is the invention of a
| purely positional system for writing numbers. Before the
| invention of such a system for writing numbers, the words meaning
| "zero" were used in all languages only for the unique quantity
| "zero". They were not used as components of the numerals used to
| name bigger numbers.
|
| The necessity to write very big numbers for accounting or
| computational purposes has made desirable the invention of a
| system that would be less cumbersome for such big numbers than
| the system used for the spoken numerals.
|
| That was the positional number system, where a small set of
| symbols is sufficient to write even very big numbers. Any purely
| positional number system needs some kind of symbol for zero,
| which must be usable in any position.
|
| When the sexagesimal Mesopotamian number system began to be used
| positionally, a symbol for "zero" had to be added. So nobody
| thought directly about a "zero" symbol. They just wanted to reuse
| the same symbols in all positions, and sooner or later someone
| understood that adding a symbol for "zero" is the solution to
| this problem.
| mensetmanusman wrote:
| Null is closer to 'empty set' than to zero, and the two
| concepts are similar but different enough to be distinct.
| dayjaby wrote:
| In German, it's the same. I hope you are not talking about
| null in coding? Because I'm pretty sure human 2000 years ago
| did not know any programming code or even empty sets.
| rdtsc wrote:
| Empty set idea doesn't seem so abstract. A bag of stuff can
| have nothing in it, so it's empty. Combining and splitting
| up containers of items would have been an everyday
| occurrence.
| kreetx wrote:
| I'm just nitpicking, but it's unlikely that the inventors of
| both had empty set and zero formalized to a degree to make
| them distinct.
| adrian_b wrote:
| While the empty set and the number "zero" are distinct
| concepts, the relationship between them is much closer than
| between any other sets and numbers.
|
| The cardinal numbers are equivalence classes of the sets.
| For any other cardinal number except "zero", the
| equivalence class of that number contains a huge number of
| sets, potentially infinite.
|
| For "zero", the equivalence class contains only a unique
| set, the empty set. Because of this one-to-one
| correspondence between the empty set and "zero", they may
| be interchanged in many contexts without causing any
| ambiguities.
| kreetx wrote:
| Right. What I was aiming at was that when someone invents
| one of these, they are probably thinking about the other,
| too, without the set theoretic rigor you recite.
| adrian_b wrote:
| In Latin, "null" was not a noun (i.e. substantive), but an
| adjective applied to nouns.
|
| It was used in precisely the same word contexts as the words
| for "one", "two", "three" etc. and in those contexts you
| could substitute any cardinal numeral. Like any cardinal
| numeral, it could be used to answer questions about how many
| things are in a certain place.
|
| So it was really the number "zero".
|
| For the empty set, the most appropriate Latin word was the
| noun "nihil" ("nothing"), sometimes contracted to "nil" (with
| long "i") hence the NIL of LISP for the empty list.
|
| So the concepts of "zero" and "empty set" were distinguished
| in Latin and also in the other known ancient languages.
|
| Some modern programming languages use "null" in a wrong way,
| when they should have stuck to the NIL of LISP. A null
| integer or floating-point number denotes the quantity "zero",
| but a null pointer is not a quantity. A null pointer points
| to nothing, so it denotes the object NIL.
| kazinator wrote:
| The word "null" in "null set" refers its _Lebesgue measure_
| being zero.
| badkitty99 wrote:
| Null sets in and of themselves have nothing to do with
| Lebesgue integrals. Null sets are definitely are not
| defined by Lebesgue integrals, you might be thinking of a
| set of measure zero which has to do with Lebesgue measure
| Someone wrote:
| > All languages have had words for "zero", for at least a few
| thousand years
|
| > What the parent article intends to discuss is the invention
| of a purely positional system for writing numbers
|
| I don't see how _positional_ is required there. Even if you
| just tally things, computations may produce zero, and you'll
| want to write something to indicate that you made the
| calculation, and didn't abandon it halfway though.
| adrian_b wrote:
| That matches in writing the meaning of the word "zero",
| "null" or whatever word is used in the spoken language.
|
| So it is not a new invention.
|
| When the positional system of writing numbers was invented,
| the symbol for "zero" had to be used in a new way.
|
| In the spoken language, one says "one hundred and five",
| without using any "zero" word.
|
| In a non-positional writing system, one would mimic the
| spoken language, writing e.g. "CV". Symbols for "0" are not
| used inside any other number, but only for the unique
| quantity "zero".
|
| In a positional system, one writes "one hundred and five" as
| "1" "0" "5", using a "0" in the appropriate positions, which
| allows the reuse of the symbol "1", instead of having to
| invent special symbols for hundreds.
|
| For very big numbers, the economy of symbols brought by the
| positional system is great, so its invention has been very
| important.
|
| The positional system did not need to invent a new word or
| symbol for "zero", but it had to invent a new way to use
| "zero", inside the strings used for writing big numbers.
| Someone wrote:
| I disagree. The Romans had the word "nulla" to (loosely)
| mean "nothing", but for them, the smallest whole number was
| 1 (https://en.wikipedia.org/wiki/Integer#History:
| _"Historically the term was used for a number that was a
| multiple of 1, or to the whole part of a mixed number. Only
| positive integers were considered, making the term
| synonymous with the natural numbers"_
|
| And yes, to introduce a positional number system, you need
| some way to indicate "there is a position here, but there's
| nothing there", but you don't need a positional number
| system to extend the integers by adding the novel concept
| of zero (and then, you need a, preferably compact, way to
| write them)
| adrian_b wrote:
| As I have said "null" does not mean "nothing". The latter
| is a substantive noun, while the former was an adjective
| in Latin, where such words were not interchangeable.
| "Nihil", i.e. "nothing", could be the answer to the
| question "What is in that box?". On the other hand, the
| question "How many eggs are in that box?" could be
| answered with "nullum ovum" or "nulla ova" ("no eggs").
| In English, "nothing" and "no", when the latter means
| "zero", are also not interchangeable.
|
| You have linked to a discussion of the word "integer",
| which has never been used by the Romans. "Integer"
| started to be used for numbers only in the late Medieval
| Latin.
|
| As I have explained, the word "null" (i.e.
| nullus/nulla/nullum) had a grammatical distribution
| identical to that of any other number, with no difference
| from 1, 2, 3 etc.
|
| Following the Greek tradition, the Roman grammarians did
| not classify their words based on their meanings or
| actual grammatical roles, but based on their kinds of
| morphological flexion (i.e. declension or conjugation).
|
| For historical reasons, the Indo-European numerals from
| one to ten had a special declension, which was different
| from the declension of any other words.
|
| Because of that, the Greeks and the Romans classified the
| numerals from one to ten and any other words derived from
| them into a special subclass of the nouns, the numerals.
|
| The words used for "zero", both in Greek and in Latin,
| are much more recent than the words for "1" to "10", so
| when they have been coined they have received the regular
| adjectival declension, instead of the archaic numeral
| declension. This difference in declension prevented the
| classification of "null" as a numeral, even if it
| belonged to a group of words about which various ancient
| grammarians have expressed doubts about how they should
| be properly classified.
|
| The ideas of the ancient grammarians about the relevance
| of declension for word classification do not matter for
| the classification of "null" from the point of view of
| modern grammar and mathematics. Even in antiquity, there
| were people like Aristotle, who have set the bases of a
| classification of the words based on meaning and
| grammatical roles, not on flexion (in the so-called
| "Categories"), even if, at least in the surviving works,
| this has not been applied to a detailed analysis of a
| language like the Ancient Greek.
|
| As it is said in the well known quotation about ducks, if
| one reads the surviving Latin texts, there is no
| difference in usage between "nullus/nulla/nullum" and any
| other cardinal number, i.e. all the cardinal numbers
| including "nullus/nulla/nullum" appear in the same word
| contexts, where they are interchangeable, therefore
| "nullus/nulla/nullum" is a cardinal number, based on how
| the Romans were using it, regardless whether Priscianus
| would have agreed to this.
| empath75 wrote:
| > "Historically the term was used for a number that was a
| multiple of 1, or to the whole part of a mixed number.
| Only positive integers were considered, making the term
| synonymous with the natural numbers"
|
| In fact, they didn't even consider 1 to be a number, let
| alone zero -- at least aristotle didn't, and his
| definition was the accepted one in Europe for many
| centuries.
|
| Though I think you need to sort of separate out what
| philosophers and mathematicians thought about numbers
| from what regular people did. I think people obviously
| had an intuitive understanding of zero and one as
| quantities, even if it wasn't formally defined that way
| by philosophers for thousands of years.
| vishnugupta wrote:
| > Even if you just tally things, computations may produce
| zero
|
| It may produce zero but not necessarily.
|
| Tallying things is more likely to produce a notion of
| "alright, no one owes anyone anything", and there's no need
| to track this "nothing". Only when things don't tally is it
| required to keep track.
|
| Because there's no use persisting this "nothing"ness over
| space and time it's unlikely to have been abstracted further
| to zero.
| rokisen157 wrote:
| Null mean nothing but not zero. In Tamil language it is called
| as Paazh. Zero is called as Suzhiyam.
| vishnugupta wrote:
| > That was the positional number system, where a small set of
| symbols is sufficient to write even very big numbers.
|
| Just the other day I was thinking about it and realised that
| positional number system enables us to represent _infinite_
| numbers with a _finite_ set of symbols. Whoever (person
| /community) invented is a genius, as positional numbering
| unlocks so many further inventions down the line.
| Chinjut wrote:
| You can already represent infinitely many numbers with a
| finite set of symbols without positional notation, in unary.
| You can write "1 + 1 + 1 + 1" or "This number of stars: ****"
| or whatever. This idea (make N marks to represent the natural
| number N) has surely been understood about as long as
| counting has been understood.
|
| The advantage of positional notation is greater efficiency at
| representing large numbers.
| vishnugupta wrote:
| Agreed.
|
| I forgot to mention that positional numbering's space
| complexity is log.
| kragen wrote:
| this is also true of clauses in grammar. the use of clauses
| makes it possible to write an infinite number of sentences
| using a finite number of words, each of which is drawn from a
| finite vocabulary, but whose assemblages are more infinite
| than the stars in the sky which so drew the attention of
| archimedes when he approximated the finite number of grains
| of sand that would be required to fill up the observable
| universe, a calculation which unfortunately he got low by
| several orders of magnitude due to an unfortunate lack of
| evidence for the hypothesis that the stars were other suns,
| not to mention the idea that some of the nebulae in the sky
| were entire galaxies of other stars, a fact which wasn't
| known until only a century ago
|
| the previous sentence contains 125 words, and it should be
| evident that it could be extended indefinitely in any known
| human language (except possibly piraha) without doing any
| violence to the rules of grammar, though perhaps great
| violence to the canons of courtesy to readers. if we use
| shannon's early estimate of 11.82 bits of entropy per english
| word, in english there are about 21478 [?] 10445 perfectly
| unremarkable sentences of that precise length, a number which
| (it should be evident) grows exponentially with the sentence
| length
|
| so, while i agree this concept is genius, it is part of the
| invention of language as we know it, and no isolated human
| tribe without language has ever been discovered. it probably
| dates to so-called behavioral modernity, at least 50000 years
| ago--probably longer than that
| feoren wrote:
| No, Log10(infinity) = infinity.
|
| It allows us to represent numbers using log symbols. Symbols
| required = O(log10(N)). Or put another way, the max number we
| can represent with S symbols is exponential in S. N =
| O(exp(S)).
| fwip wrote:
| They are speaking of a finite set of symbols (the digits 0
| through 9), not the number of glyphs in the representation
| of an arbitrary number.
| kragen wrote:
| > _When the sexagesimal Mesopotamian number system began to be
| used positionally, a symbol for "zero" had to be added_
|
| the sexagesimal mesopotamian number system was used
| positionally from the beginning (that's what was sexagesimal
| about it) and, as the article explains, did not have a symbol
| for zero in common use, or evidently at all for millennia. the
| early zero symbol mentioned in the article is from a tablet
| from 0700 bce or later, which is only 2700 years ago. at that
| point positional sexagesimal mesopotamian numerals were already
| about 1000-2000 years old. what popova doesn't explain is that
| they often used an empty space for zero instead
| constantcrying wrote:
| I am very unconvinced by this "history of zero". Definitely the
| Greek geometers were aware of that concept, they just expressed
| it geometrically not numerically.
|
| Putting the concept at some specific geographic location seems
| very strange to me. To me there is no doubt that in each
| mathematical culture there was some notion of it, just weaved
| into that conceptions of that particular culture.
|
| Of course then there is zero as a symbol and positional number
| systems. The first one seems very uninteresting, the later one
| definitely more so, but the question of how to express numbers is
| definitely more interesting and broad than the history of just
| one component of it.
| rusticpenn wrote:
| Mesopotamians were at least a millennium ahead of Greeks
| though..
| constantcrying wrote:
| No. They weren't even in the same category, even the
| comparison doesn't make sense. In Mesopotamia mathematics
| seems to have been a tool, a method for business and
| construction planers to drive certain quantities.
|
| Mathematics to the Greeks was "mathematics" in the sense we
| understand it today. From a system of axioms they derived a
| complex system of theorems, which allowed for an abstract
| description of reality. It was mathematics as a system of
| truth, which then could be used for other purposes. E.g. with
| Archimedes who discovered integration, but also was a
| prolific engineer.
|
| The Greeks were so far ahead of anything the Mesopotamians
| did, that even the comparison is unfair.
| derstander wrote:
| > No. They weren't even in the same category, even the
| comparison doesn't make sense.
|
| The comparison does make sense if you interpret the
| parent's post simply to state that Babylonian mathematics
| (in Mesopotamia) were developed (and seemingly stagnated)
| before Greek mathematics began in earnest. Which seems to
| be pretty uncontroversial. There are extant clay tablets
| from 1800 to 1600 BC that would indeed predate the Greek
| Geometers by a millennia -- and that's if you're counting
| Thales as the beginning.
|
| Ie. your parent post is using "ahead of" == "temporally
| before" not as in "more advanced".
| rusticpenn wrote:
| Thank you.
| constantcrying wrote:
| Are you really thanking someone for defending your stupid
| gotcha?
|
| What are you even trying to do here? Show me that ancient
| Mesopotamia had mathematics? Wow! What a stunning
| revelation, genuinely brilliant insight from you.
| constantcrying wrote:
| No. You are still ignoring that the mathematics of the
| Greeks and the mathematics of the Mesopotamians were
| radically different. Pointing out that at some earlier
| point the Mesopotamians had some form of mathematics is
| totally irrelevant to the point.
|
| Did you even read my original post? My position is that a
| linear history of the concept of 0, which spreads from
| India to Mesopotamia to Greece is nonsense. That the
| Mesopotamians did some mathematics long before the Greeks
| did is _irrelevant_ to my argument. It totally does not
| matter. What makes it even more ridiculous is that my OP
| obviously assumes that the Mesopotamians had mathematics
| before the Greeks, so pointing it out as some "gotcha"
| is just really stupid.
|
| It genuinely makes me mad to have this low quality
| discussion, where someone barges in and gives you a
| trivial "gotcha" as if I wasn't completely aware of that
| fact. And when you try to point out why the person didn't
| understand what you were saying you are getting another
| person debating the stupid gotcha, as if it even
| mattered.
| rusticpenn wrote:
| I am of the opinion that there is something to learn from
| everyone. However I come to HN for civil discussion. That
| was not a gotcha or are we trying to win something.
| Mesopotamians needed complex mathematics for taxes long
| before even Mycenian Greeks came into picture. You are
| talking about classical Greeks. Naturally there were
| advancements in mathematics during the Greek period, but
| that does not undermine the achievements of
| Mesopotamians.
| constantcrying wrote:
| Your response was a single sentence, pointing out
| something which I obviously knew. That information was
| totally irrelevant to what I said, but you thought you
| had discovered something which I didn't know about. You
| also obviously didn't understand what I said, else you
| wouldn't have posted a single statement with no new
| information. Obviously this was a gotcha. You made no
| attempt to even try to understand me or post anything
| interesting or relevant, instead you took a cheap shot at
| a misreading of my post.
|
| If this is what "high quality discussion" is to you, what
| does mid level discussion look to you? Just a slur as a
| response?
|
| >Mesopotamians needed complex mathematics for taxes long
| before even Mycenian Greeks came into picture. You are
| talking about classical Greeks. Naturally there were
| advancements in mathematics during the Greek period, but
| that does not undermine the achievements of
| Mesopotamians.
|
| I am stunned by your knowledge. You really know that?
| That is breathtaking information. Genuinely thank you for
| posting this novel information that I was totally unaware
| of. Bringing that to my attention was extremely relevant
| and has radically changed my position.
|
| Seriously fuck off, you still haven't even bothered to
| read my OP.
| rusticpenn wrote:
| I am sorry anonymous person on the internet. I have no
| idea what you know or do not know.
|
| I am perplexed by the notion that time does not play a
| role for you when talking about the "history" of
| something.
|
| Just like the fact the Phoenecian alphabet was modified
| by the Greeks, there would have definitely been transfer
| of knowledge.
| constantcrying wrote:
| You are just impossible to talk to. Why are you _still_
| posting random gotchas at me?
| hifromwork wrote:
| I have no intention to argue with either of you, but from
| an outside perspective, it's you who is very hard to
| discuss with.
|
| I don't understand what's wrong with rusticpenn original
| post and I don't think they argue in bad faith. Maybe you
| are right, but because of all the personal attacks you
| make, it's hard to see your point, and you also come of
| as very rude. I don't write this to criticize you - I'm
| just saying that if you change the way you argue, you
| will probably convince many more people to your ideas.
| Constructive discussion is, in general, a very useful
| skill in life.
| kragen wrote:
| maybe try to have a little more patience with other
| people; they may not find it as easy to understand you as
| you think it is, and it doesn't always mean they aren't
| trying. sometimes they're just dumb, sometimes what you
| wrote is actually pretty unclear (even though it's
| perfectly clear to _you_ because you already know what
| you were trying to express), and sometimes they 're
| actually aware of things you're not which introduce
| hidden contradictions into your narrative
|
| the worst case, from my point of view, is when they're
| actually just dumb and you yell at them for their
| intellectual limitations. that's sad
| derstander wrote:
| > No. You are still ignoring that the mathematics of the
| Greeks and the mathematics of the Mesopotamians were
| radically different.
|
| This doesn't matter. We're talking about 0. Whether
| mathematics are conceptualized as a tool or an academic
| discipline in its own rite wasn't the thrust of the
| article. It wasn't even part of your original post.
|
| > Did you even read my original post?
|
| It might surprise you, but yes -- yes, I did!
|
| > My position is that a linear history of the concept of
| 0, which spreads from India to Mesopotamia to Greece is
| nonsense. That the Mesopotamians did some mathematics
| long before the Greeks did is irrelevant to my argument.
| It totally does not matter. What makes it even more
| ridiculous is that my OP obviously assumes that the
| Mesopotamians had mathematics before the Greeks, so
| pointing it out as some "gotcha" is just really stupid.
|
| This is a better statement of your position than you
| originally posted, when you said "I am very unconvinced
| by this "history of zero". Definitely the Greek geometers
| were aware of that concept, they just expressed it
| geometrically not numerically." Here's why I think your
| initial statement is a weak argument for your refutation
| of a linear history of the concept of zero: Babylonian
| mathematics temporally happened first _and_ the general
| opinion (when I was in school, at least) was that
| Babylonian mathematics likely influenced Greek
| mathematics. That 's why I think this is a weak argument
| to argue against the linear history concept of 0.
|
| > It genuinely makes me mad to have this low quality
| discussion, where someone barges in and gives you a
| trivial "gotcha" as if I wasn't completely aware of that
| fact. And when you try to point out why the person didn't
| understand what you were saying you are getting another
| person debating the stupid gotcha, as if it even
| mattered.
|
| I'm sorry you're mad but I mostly disagree on your
| assessment of the quality of the discussion. Your top-
| level comment on the article about Mesopotamia creating
| nought was essentially that Greek geometers knew about it
| and that tying concepts and geographical locations seems
| odd.
|
| I don't disagree with your second point, but we may be in
| the minority: lots of people are very interested in
| knowing who had what ideas first and where. Your first
| point doesn't stand well alone without the further
| elaboration you've made through the rest of this
| discussion. With the elaborations you've made since the
| rest of this thread might not have happened.
| constantcrying wrote:
| I wanted to give a seriously reply, but clearly there is
| no point.
|
| I still don't get why you and the other guy think
| pointing out that "Mesopotamians did it first" is
| relevant. Even if they did and even if the Greeks were
| extremely influenced by them, their concept of a
| geometrical zero was still radically different to the
| Mesopotamians notion, so it is totally irrelevant who was
| first. The only counter argument _could_ be that they had
| the same notion, which the Greeks adopted from the
| Mesopotamians as a complete package.
|
| I don't think this is really complicated. If their
| understanding was radically different, then the concept
| couldn't have just "moved over", so disprove my thesis he
| discussion on "who was first", is obviously irrelevant.
|
| > With the elaborations you've made since the rest of
| this thread might not have happened.
|
| Well, the genuine curiosity and willingness to discuss
| the subject embodied in a single sentence dismissing my
| post because of an obviously true statement made me very
| glad to elaborate and discuss further.
| derstander wrote:
| I'm going to step back to a meta level and make a couple
| points:
|
| - Several times in the context of this discussion you've
| made negative assumptions about someone or their motives
| replying to you (e.g. "I wanted to give a seriously
| reply, but clearly there is no point" and "What a
| stunning revelation, genuinely brilliant insight from
| you"); I don't think this fosters good discussion (and I
| also don't think it's particularly good for mental
| health)
|
| - You seem particularly aggrieved by someone not
| automatically assuming you know something. Yes, we all
| understand now that you know the Mesopotamians predated
| the Greeks with regard to the mathematical notions we're
| discussing here. But you know what? _There 's nothing
| wrong with NOT knowing_. And some people reading the
| discussion might not have known.
|
| Now back to the main topic at hand.
|
| > Even if they did and even if the Greeks were extremely
| influenced by them, their concept of a geometrical zero
| was still radically different to the Mesopotamians
| notion, so it is totally irrelevant who was first
|
| Now I _finally_ feel as though I understand the main
| thrust of your argument. Let me restate your hypothesis
| to see if I 've got it: even though the Babylonians had
| the concept of zero, it was sufficiently lacking compared
| to the Greek concept as to be incomparable; the Greeks
| fundamentally independently invented the concept of a
| zero more mathematically powerful than the Babylonians
| rather than took the existing Babylonian concept of
| nought and developed it further. Is that right?
| kragen wrote:
| > _from India to Mesopotamia to Greece_
|
| surely nobody thinks that zero spread from aryabhata 1500
| years ago to the mesopotamians+ 2700 years ago, and
| classical and hellenistic greece didn't even have a
| numeral for zero, much less the idea that zero was a
| number. (obviously the greeks have been using a numeral
| for zero for several hundred years now)
|
| the hypothetico-deductive method that defines what we
| call 'mathematics' today does seem to have originated in
| the hellenistic period, although probably in alexandria
| rather than in greece proper. but menon and the timaios
| show that the pythagoreans already had the rudiments of
| it in classical greece, and there's no evidence that
| anything similar existed during the thousand-plus
| preceding years of mesopotamian mathematics. i think
| that's what you're saying
|
| on the other hand, the majority of greek mathematics
| during the classical and hellenistic period was still
| mostly 'a tool, a method for business and construction
| [planners] to [derive] certain quantities', even if the
| calculations were being done on sand-tables instead of
| counting-boards. systems of axioms don't make an
| appearance in the surviving written record until roughly
| euclid; aristotle, plato, and pythagoras evidently
| weren't yet familiar with what we call 'mathematics' and
| in particular 'proof', although the later mathematicians,
| especially in the hellenistic period, did build on their
| work
|
| so i think it's a mistake to talk about 'the greeks' as
| if they were a single person, like in a game of civ; all
| the greeks alive in plato's time were already dead by
| euclid's time, euclid didn't even live in greece, all the
| greeks alive in euclid's time were already dead by
| ptolemy's time, there are still greeks today, and in any
| time period, different greek people knew different things
|
| ______
|
| + https://www.youtube.com/watch?v=jAMRTGv82Zo
| kouru225 wrote:
| This is the difference between philosophy and number
| systems. The Babylonian number system was, by far, the
| superior system to the Greek number system (which they
| inherited from the Egyptians.) This is something even the
| Greeks acknowledged because they would use the Babylonian
| system to do their own calculations, and then convert it
| back into the Greek numbers (similar to how the USA
| scientists use the metric system and then convert back to
| the imperial system today).
| empath75 wrote:
| The question is when they used it or understood it _as a
| number_. The greeks didn't even consider _1_ to be a number,
| since "number" implied a "multitude". It was controversial as
| to whether 1 was a number up until the 1400s or 1500s, let
| alone zero.
| taeric wrote:
| This feels wrong, at face value. Even in modern English, if I
| say I have a number of things, I almost certainly don't have
| just one. This does /not/ mean that one is not a number. It
| /does/ mean that the word "number" can have several uses.
| empath75 wrote:
| more here: https://plato.stanford.edu/entries/aristotle-
| mathematics/#10
| taeric wrote:
| But this doesn't contend with my point? A sibling point
| brought up confusion of cardinal and ordinal numbers. In
| modern english, I challenge you to find a good
| understanding outside of advanced practitioners on the
| difference.
|
| I think it is fair to say that my statement is too strong
| to say it is wrong. My assertion would be that it is more
| complicated and almost certainly there is a lot lost in
| translation along the years.
| wcarey wrote:
| Euclid distinguishes the unit (VII, Def.1) and number -- a
| multitude composed of units -- (VII, Def.2). His definition
| of prime and composite numbers (VII, Def.11 and Def.13)
| clearly exclude one from the group of numbers, otherwise
| every number would be composite.
| taeric wrote:
| I think this is good to argue that my assertion is likely
| too strong. I fear this is close to arguing that early
| programmers were not familiar with map/flatMap. They did
| not discuss it as a first class thing, sure. Was it
| completely alien to all practitioners? I find that harder
| to swallow and it is likely that we are debating methods
| versus functions completely removed from the context in
| which the words were largely used.
| kragen wrote:
| thank you! i was wondering about that
| meroes wrote:
| Wasn't 2 the smallest actual number to Greeks? I think it's
| clear from the history Greeks did a TON of math but missed this
| number system and abstraction. And if you say they still
| understood zero but didn't have number system to tag along
| (they didn't have Hindu Arabic numerals) well a better number
| system provides so much more math that the argument doesn't
| have much oomph. There's no doubt the Hindu Arabic numeral
| system, of which zero was apart, was the mathematical
| development bar none of its time. Did anyone really understand
| zero before it could do all these useful things in equations?
| Doubtful. And that's the most charitable take for the Greeks
| regarding zero.
| kouru225 wrote:
| I'm in the process of reading this article, but I recently read
| Zero: The Biography of a Dangerous Idea by Charles Seife (which
| is a great book) so I feel like I can explain:
|
| The Greeks were ideologically opposed to the number zero.
| Aristotle outright refuses to acknowledge the existence of zero
| and of infinity. The Greeks were aware of the idea of zero, and
| they even used zero when they calculated using the Babylonian
| number system (which used zero as a placeholder number,) but
| they always converted the numbers back into their own system,
| and stubbornly refused to acknowledge its existence.
|
| The fact that the Greeks saw geometry and math as
| interchangeable was their weakness here. There's no way to
| represent the number 0 geometrically, but the Greeks weren't
| gonna give up their belief in Geometry because it provided them
| with social and political power.
|
| Pythagoras and Aristotle believed in a religious philosophy
| with logos at the center. Logos can be translated as "thought"
| or "word" (as it is in the Bible) or it can be translated as
| "ratio." This is because they saw all these things as one (the
| Latin translation of the Greek word "logos" is "ratio.") The
| ratio of numbers was thought the be the underlying mechanism
| that proved the order of the universe (which naturally saw the
| nobility as orderly and the peasantry as chaotic). This was a
| profoundly powerful sociopolitical tool that ended up spreading
| all across the world because Aristotles student just happened
| to be the greatest conqueror of the era: Alexander The Great.
|
| Anything that threatened the philosophy of logos was
| suppressed, violently. Hippassus and Zeno were both murdered
| for the crime of talking about irrational numbers and infinity.
| Zero was one of these threats. 1:0 = infinity, 10:0 = infinity,
| anything:0 = infinity. This was not logos and therefore it was
| suppressed.
|
| This philosophy extended beyond mathematics into the realm of
| astronomy and, weirdly enough, music (at the time, Pythagoras
| was actually most famous for his discovery of the golden ratio
| using an instrument called the monochord, which is a legend
| that seems to be false but nonetheless made him very famous.)
| This astronomical belief system was then later attributed to
| Ptolemy. This philosophy then was transplanted into Christian
| theology, and it took centuries for the monks to accept the
| existence of zero and infinity as a result. We even have cases
| of religious figures persecuting mathematicians about zero and
| infinity as late as the 1800s.
| plasticeagle wrote:
| Zero is important because it is the first abstraction. If your
| notion of numbers includes the idea of zero _as a number_ , then
| you have broken through the first intellectual barrier of
| mathematics. Without this, it remains tempting to still consider
| numbers as existing for the purpose of counting _things_.
|
| But with zero, this idea converges on the same thing. No matter
| what things you were counting, if you have zero of them, you have
| the same idea. And so you take a step towards the idea of a
| number being a concept in its own right, rather than existing
| purely for the purpose of counting or measurement.
|
| It is the same sort of conceptual freedom that allows you to do
| things like add a number to a square. To deal with an equation
| like x + x ^ 2 = 0. If you're stuck with numbers "meaning"
| something beyond themselves, then you'll never add x to x^2. One
| is a length, the other an area. They are different objects.
|
| This intellectual leap is one that must be made by all students
| of mathematics - and many young people do not.
| traject_ wrote:
| Yeah, a lot of these articles conflate use of zero as
| placeholder, numeral and number. But the real critical
| conceptual step is the last step of using zero like any other
| number (mostly) in arithmetic.
| roywiggins wrote:
| I am pretty sure early European mathematicians did treat
| polynomials as areas or volumes, x can be just a rectangle x
| units long in one dimension and 1 in the other, x^2 is a
| square, etc. This meant they had to go through contortions to
| avoid negative coefficients since they made no geometric sense.
| If a coefficient would otherwise be negative it would have to
| move to the other side of the equality, and be solved using a
| different method. Instead of a single quadratic formula they
| needed several different cases depending the exact form of the
| polynomial.
| dahart wrote:
| > Zero is important because it is the first abstraction
|
| I'm not sure what this even means, but Sumerians had abstract
| mathematics, in addition to art and literature which are
| abstract by their nature. They were using numbers in the
| abstract sense before the number zero was named, and so while
| it seems like a logical and tempting narrative that naming zero
| is what abstracted numbers, history doesn't seem to support
| this particular post-facto rationalization. Naming zero is very
| important in the history of math, it just isn't the first
| abstraction.
|
| https://en.wikipedia.org/wiki/Sumer#Mathematics
| plasticeagle wrote:
| You may very well be entirely right.
|
| And I think you know exactly what I meant, because you
| immediately countered with some historical evidence around
| those first abstractions.
|
| So, thank you, and I will read much more about Sumerian
| mathematics with great interest.
| adolph wrote:
| > Zero is important because it is the first abstraction
|
| And thus was born the everlasting confusion between cardinal
| and ordinal numbers.
| kragen wrote:
| regardless of whether it makes sense to say that treating zero
| as a number was 'the first abstraction' or not+, ancient
| mesopotamians did not treat zero as a number. even noble
| fibonacci didn't consider it as a number or even a digit:
|
| > _The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these
| nine figures, and with the sign 0 (...) any number may be
| written._
|
| i'm not sure when it became conventional to consider it a
| number rather than the absence of one; it might not have been
| until the early modern era
|
| https://news.ycombinator.com/item?id=40917674 suggests that
| european mathematicians still hadn't agreed that 1 was a number
| until the early modern era
|
| ______
|
| + it doesn't
| meroes wrote:
| Why is that the first abstraction? I would think at least one
| of these three earlier developments would be:
|
| 1) deductive geometry of Thales, where we could now prove
| things independent of the physical world (abstracting away from
| the physical)
|
| 2) Plato's remarks on incommensurability (proto irrational
| numbers) being something real but not physical, because no
| physical process could prove to the mathematician that two
| lengths really have no common unit measure. Here the
| abstraction is again away from physical means.
|
| 3) infinity of numbers. Abstracting away from large but finite
| collections. We can only ever survey finite collections
| physically, again is an abstraction.
| njrc wrote:
| > Zero is important because it is the first abstraction.
|
| Aren't numbers themselves an abstraction?
| plasticeagle wrote:
| Initially, no. They are a mechanism for counting, or a way of
| recording measurements. Many young people, when trying to
| learn mathematics, do not take the necessary step beyond
| these ideas.
|
| They are not helped by educators insisting on presenting
| "word problems" when teaching maths. To get to the next
| level, you need to break the connection between numbers and
| the "real world". I've always felt that the number zero
| represented the first step that humanity took in this journey
| - and it's a step that every human also needs to individually
| take if they want to learn maths.
| chrisweekly wrote:
| Two things:
|
| 1. I'm a huge fan and long-time supporter of Maria Popova; IMHO
| themarginalian.org is one of the finest websites ever, with a
| breadth and depth of consistently worthwhile content. Check it
| out!
|
| 2. This post reminds me of a book I really enjoyed maybe 10 years
| ago -- "Zero: the History of a Dangerous Idea". Recommended.
| nyc111 wrote:
| "This concept of the infinite in a sense contoured the need for
| naming its mirror-image counterpart: nothingness."
|
| Is infinity the mirror image of nothingness? I cannot conceive
| infinity let alone its mirror image. Same with nothingness.
| ArnoVW wrote:
| If you have young kids that you would like to infect with the
| virus of mathematics, I highly recommend "the story of 1" by
| Terry Jones (of Monthy Python fame)
|
| They cover how we went from objects to abstract concepts, and how
| 1 and 0 were "invented".
|
| https://www.youtube.com/watch?v=1Tm18iapVlI
| osigurdson wrote:
| I don't understand how the concept of zero could have taken so
| long for humans to discover: "I had two apples, I ate one and
| gave one to a friend". They must have had a word for the concept
| of running out of things as soon as any level of cognitive
| ability emerged.
| dahart wrote:
| That's right; the concept of nothing was already there, both as
| an idea, as well as in written language. It's the use of a
| written numeral zero for arithmetic, and the implications of
| using zeroes in algebra that this article is discussing.
| xigoi wrote:
| What took long to find out is that zero is a number like any
| other. That having no apples is just a special case of having
| some quantity of apples, not something completely separate.
| ks2048 wrote:
| Shoutout to the Maya who also had zero (as a placeholder in base
| 20 number system), it seems by at least 36 BC
| kouru225 wrote:
| I just finished Zero: A Biography of a Dangerous Idea.
|
| One of the best books I've read in a while. Really gives a great
| story about the evolving history of thought.
| gumby wrote:
| This article is really just a gloss on Kaplan's book (which I
| have read so it stands out) with a bit of gratuitous randomness
| thrown in.
|
| Much as I have a rather low-level atavistic desire to credit
| India with the zero/nil, as there was so much exchange between
| mesopotamia and the early indus regions (just look at the idea of
| the alphabet going one way and then digits going the other), the
| sumerian "origins" are quite likely. More importantly, the vedic
| tradition didn't give rise to the formalisms developed by the
| later Greeks and, centuries later, their Islamic students. Thus
| for a long time, scholarly dissertations from the subcontinent on
| mathematics, philosophy etc tended to essays and explorations of
| conjectures, which makes pinning responsibility hard to do, the
| way you can, say, "Wiles did prove Fermat's Last Theorem".
|
| Personally I find invention requires so much history and
| intertwined communication that the idea of "inventor" is kind of
| bogus anyway.
|
| BTW in case this sounds like I'm dissing ancient indian scholars:
| you see this in the early days of any scientific field: early
| neuroscience in the early 20th century, the same with cognitive
| science in the mid 50s-70s (at least) etc. In fact most of
| contemporary ML just has a light layer of formalism painted on
| too. It feels like fields need names, but only really get them
| when they have attained some early level of abstraction and
| emerging rigor.
|
| Sorry, that moved on beyond zero!
| jacobolus wrote:
| The real problem is there are too many people who treat
| mathematical/scientific history as some kind of olympic games,
| where the goal is to rack up medals for your preferred team
| (country, region, religious group, ...), with the result that
| any discussion becomes counterproductively politicized.
|
| Trying to argue about whether one tribe's or another tribe's
| 50-generations-ago ancestor was the first one to do this or
| that thing seems to me like completely missing the point, when
| all of these steps were part of a long and gradual historical
| process, building ideas and tools up over centuries.
| (Similarly, it's annoying how many debates center on various
| ancient figures' ethnicity or religious affiliation, usually
| without much evidence.)
|
| To anyone who tries researching ancient (or more recent)
| mathematics, it's clear that there usually isn't a single aha
| moment changing everything, but a broader culture that
| gradually evolves. We can see different flavors/aspects of a
| concept like "zero" which were developed different times and
| places (China, Mesopotamia, India, Greece, North America), none
| of which really draws any obvious line in the sand.
|
| With regard to Indian innovations, however, it seems pretty
| clear that written arithmetic per se (performed on a "sand
| board") was developed there, as credited by all of the oldest
| extant texts on the subject from writers in Arabic (which call
| it something like "Indian arithmetic" or "Indian numbers").
| Written arithmetic was then substantially elaborated in the
| Islamic world with a switch to using pen and paper, before
| making its way to Europe where it eventually kicked off the
| development of modern mathematical notation. The earlier
| Mesopotamian/Egyptian/Greek/European tradition, as well as the
| Chinese tradition, were generally based on using finger
| counting or some form of counting board, with written numerals
| used as a serialization format rather than a calculation tool.
| Arguably the invention and spread of physical materials like
| cheap good quality paper, writing implements, ink, and
| eventually printing presses were as important as the
| theoretical developments.
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