[HN Gopher] How the human brain contends with the strangeness of...
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How the human brain contends with the strangeness of zero
Author : apollinaire
Score : 60 points
Date : 2024-10-26 06:22 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| ofrzeta wrote:
| "That's because to understand zero, our mind must create
| something out of nothing. It must recognize absence as a
| mathematical object." - I don't see how that follows from
| anything. So 0 in counting would account for "no item", while 0
| in the decimal system would account for "multiply by 10"? Are
| there different neurons for that? :)
|
| I've also read the linked article
| "https://www.quantamagazine.org/why-the-human-brain-perceives...
| which discusses the discovery of "number neurons" and goes on to
| discuss that there might be no "number neurons" after all.
| t-3 wrote:
| I feel like zero in the context of counting is very simple and
| intuitive, and in addition, subtraction, and multiplication as
| well (aren't those just extensions of counting anyway?). In
| division and exponentiation things can be a bit harder to
| comprehend - why is dividing by zero impossible? Why is the
| zeroth power always one? I reckon the barrier to understanding
| there is that division and exponentiation are not counting
| operations but meta-operations relating to multiplication.
| jon_richards wrote:
| Exponentiation to zero annoyed me as a kid because I had
| equated "no operations" as zero from the pattern 2x2=2+2,
| 2x1=2, 2x0=
|
| Declaring 2x2=0+2+2, 2x1=0+2, 2x0=0 while 2xx2=1x2x2,
| 2xx1=1x2, 2xx0=1 seemed arbitrary.
|
| What helped was learning about negative exponentiation and
| exponentiation simplification, so 2xx0 = 2xx2 x 2xx(-2) =
| 2xx2 / 2xx2 = 1.
|
| That said, I clearly still take issue with unintuitive
| interpretations of "nothing"
| https://stackoverflow.com/questions/852414/how-to-
| dynamicall...
| anon291 wrote:
| I mean the reality is that in the same way that 2 * 0 is
| ONE.
|
| Or if I were using latex: 2 \times 0 =
| 1_{+}
|
| Because multiplication, being repeated addition and
| exponentiation repeated multiplication both behave the same
| way. When asked to repeat the operation zero times, they
| return the unit or 1 of the underlying group, which is
| typically denoted as 1_{whatever}
|
| However, the 1 of addition is 0 when using the standard
| notation for integers
| bmacho wrote:
| > Exponentiation to zero annoyed me as a kid
|
| Your teacher didn't tell you (or told you, but you didn't
| recognize it as something valuable and forgot it) that
| exponentiation to zero is a new definition over
| exponentiation to positive integers. Exponentiation to
| positive integers is defined somehow, and that definition
| says nothing about exponentiation to zero. It is a new
| definition, not something that you deduce.
|
| The same holds for 0^0 or 0/0 (with some amounts of
| confusion, lies, and hypocriticism).
| jon_richards wrote:
| Declaring it as a new definition _was_ what annoyed me...
|
| Which I found incredibly silly once I leaned about
| negative exponentiation and could now deduce the pattern.
|
| Similarly, 0/0 became much more tractable to me once I
| learned L'Hopital's rule
| https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule
| n_plus_1_acc wrote:
| Now, you probably now that this is because 0 and 1 are the
| identity elements of the additive and multiplicative groups
| of real numbers, respectively. Probably wouldn't've helped
| you as kid though
| jon_richards wrote:
| I think it's much more productive to teach children
| 2x0 = 2x1 + 2x(-1) = 2-2 = 0
|
| and 2xx0 = 2xx1 x 2xx(-1) = 2/2 = 1
|
| Inverting the concept and having those patterns stem
| _from_ a fundamental identity can wait until its not seen
| as the mathematical equivalent of "Because I said so".
| whatshisface wrote:
| Another good way of thinking about it is that if 10^a 10^b
| = 10^(a+b), you can set a=3 and b=0 to get 10^3 10^0 =
| 10^3. If you divide both sides by 10^3, 10^0 = 1. QED.
| anon291 wrote:
| Dividing things by zero is not impossible for some definition
| of impossible. In the sense that you can always make progress
| in pulling a group of zero from a group of something and
| increasing your counter, you can be very productive when
| dividing by zero.
|
| This has to do with rings and the relationship between the
| two identities of the two underlying groups. It ultimately
| stems from the distributive property between multiplication
| and addition
| Ferret7446 wrote:
| Dividing by zero is impossible because it's not defined.
| Simple as that. Same as undefined behavior in any other
| language or system. It's like playing chess and saying "but
| what if I lay my pawn sideways and place it on top of my
| head? What happens then?". It's not defined in the rules.
| bubblyworld wrote:
| I think this point of view is a little bit backwards - it's
| true that it's not defined (for the reals), but I don't
| think that's a convincing _causal_ explanation.
|
| I think we invent abstractions because they allow us to
| reason about patterns in reality in a consistent way. The
| fact that division by zero is undefined is (to my mind)
| because it doesn't correspond to any useful pattern in the
| parts of reality we typically apply arithmetic to
| (accounting, estimation, etc).
|
| What I like about this point of view is that it encourages
| thinking about what you are trying to accomplish, rather
| than fixating on formal rules. In some contexts "division
| by zero" can correspond to a meaningful pattern - look up
| the geometry of the projective line, for instance. In such
| cases you might want to _include_ it in your model, rather
| than declaring it to be undefined as a convention!
| Ferret7446 wrote:
| If you want to go further, it's because division is
| defined as the inverse of multiplication. It doesn't
| really have anything to do with:
|
| > it doesn't correspond to any useful pattern in the
| parts of reality we typically apply arithmetic to
| (accounting, estimation, etc).
|
| You can of course re-define it, but then we aren't
| talking about the same thing any more. The operation of
| inverting multiplication, is not defined for zero.
| bubblyworld wrote:
| But this is my point - the definitions are the way they
| are _because_ it 's a good model of counting (and other
| counting-like things). They weren't invented in a vacuum.
|
| Similarly the fact that multiplication and division are
| inverses is a _property_ of this model. Conceptually you
| can imagine splitting and copying groups of objects quite
| independently of one another (and which one you view as
| fundamental is really a post-hoc choice).
|
| In general these days we mostly see clean mathematical
| abstractions because all the scaffolding has already been
| removed by mathematicians past. And as a result people
| come to believe that this is how mathematics is done. But
| always there is an initial period of exploration (which
| eventually gets forgotten) as people try to work out how
| to axiomatise the various systems they are interested in.
| ggu7hgfk8j wrote:
| At the end of the day it's about practicality.
|
| Contrast 0^0 with 0/0. Both of them are considered
| undefined, but the former is often defined "locally", e.g
| in a given textbook or article to have the value 1.
| That's not true for 0/0 because it's not been found to be
| useful.
| bubblyworld wrote:
| Indeed, 0^0 is often defined to be 1 in combinatorics
| textbooks, for instance, because in those contexts that's
| a useful/meaningful choice.
|
| In measure theory it's often useful to augment your reals
| with positive/negative infinity. In projective geometry
| it's sometimes meaningful to allow division by zero (to
| counter one detail of your comment). In nonstandard
| analysis you would consider infinitesmals to be valid
| numbers, and in game theory you might consider stuff like
| the surreals which are yet another to view the familiar
| numbers, with different laws.
|
| You can say we're talking about many different systems
| here, and that's true in a formal sense. I'm just
| pointing out that these formal systems come from
| somewhere, and mathematics is really about the thought-
| stuff underlying them. You should be willing to bend a
| rule here and there if it is truer to the concepts you
| care about - statements like "division by zero is
| undefined" should never be taken as absolutes.
|
| (but of course this is just my personal philosophy of
| mathematics, take it with a pinch of salt)
| fph wrote:
| There definitely are contexts in which it makes sense to
| define 0/0 = 0. Example: given a vector x and an
| approximation y to it, you wish to compute the
| "componentwise error" delta, i.e., the minimum positive
| number delta such that each entry of y is within relative
| error at most delta from the corresponding entry of x:
|
| (1) abs(y_i - x_i) <= delta * abs(x_i) for each i.
|
| This number can be computed as
|
| (2) delta = max (abs(y_i - x_i) / abs(x_i)) over each i.
|
| If you wish to allow zero entries in your vectors while
| keeping the equivalence between (1) and (2), you have to
| define 0 / 0 = 0.
| gls2ro wrote:
| Diving by zero is not impossible because it is not defined,
| but because it grows ad infinitum.
|
| Take out a scientific calculator and start doing division
| by smaller and smaller units:
|
| 1/0.9
|
| 1/0.09
|
| 1/0.009
|
| ...
|
| 1/0.000000000000000000009
|
| So we can then ask what happens if we do this more and more
| with incremental steps? It seems that we go toward
| infinity.
|
| thus the more you do this the higher the returned number
| and I think we just discovered (I think I have to refresh
| my memory and hope not say something very wrong) the
| concept of calculus. Then we can use a new concept: limits
| - what happens when that 0.000000000............N is as
| close to zero as possible.
| dpassens wrote:
| Except it doesn't necessarily go towards (positive)
| infinity. Use a negative dividend, and you approach
| negative infinity instead. The same is obviously true if
| you use a negative divisor instead, and you get positive
| infinity again if both are negative. So division by zero
| is impossible because you can't meaningfully define it
| (in the general case).
| waffletower wrote:
| The strangeness of zero seems so strange to me now, though there
| must be some truth to it given the relatively recent historical
| emergence of 'zero'. Mapping mathematics to concrete objects via
| counting is an important mathematical primitive, but the concept
| of zero pervades so many other common physical phenomena --
| pendulums are a very effective case in point for both zero and
| negative numbers. Having zero of certain necessities (food,
| water) are incredibly salient and readily understood conceptually
| to young minds. It seems to me that its historical absence is
| actually exaggerated, as if zero was instead a known, but
| unspoken truth.
| praptak wrote:
| It's not only about having a concept for absence though. Zero
| provides a way to express both the absence of grain and having
| some grain in a unified way, a number.
| andsoitis wrote:
| > zero pervades so many other common physical phenomena
|
| Zero was invented approximately a thousand years before the
| pendulum was scientifically developed for use in technology
| like clocks.
| red75prime wrote:
| Absence of something surely was a known concept. Zero..., that
| is reified absence by itself with no relation to a thing being
| absent, most likely not.
| DeathArrow wrote:
| Not all mathematical objects have correspondents in the natural
| world.
|
| 0 it's a mathematical convention. We can't have 0 apples in a
| basket. We either have some apples or we don't have any.
|
| If 0 is weird, negative numbers are also weird. Infinity is
| weird. Spaces with more than three dimensions are weird.
|
| Complex numbers are weird from an algebraic perspective but not
| that weird from a geometric perspective.
|
| Empty set is weird.
|
| Everything that's abstract is weird.
|
| For my mind abstract algebra is more weird than calculus and
| mathematical analysis so that's why I enjoy calculus and
| mathematical analysis more.
|
| But being weird and abstract can also be useful.
| bmacho wrote:
| > We can't have 0 apples in a basket.
|
| I just checked my basket, it contains 0 apples.
| ulfbert_inc wrote:
| you are imagining things, basket is empty
| ykonstant wrote:
| We're imagining things either way.
| irrational wrote:
| > It's like an extra level of abstraction away from the world
| around you
|
| Is it? I ate all the apples. There are no more apples left. There
| are zero apples. Don't we come face to face with nothing left all
| the time? There are zero guests left. There are zero episodes of
| the show to watch. There are zero days until Christmas. There is
| zero money in my wallet.
| ulfbert_inc wrote:
| >There are no more apples left. There are zero apples.
|
| These are not equivalent statements though. You simply learned
| at some early age zero represents absence, but that is not a
| natural concept.
| pflenker wrote:
| To add to that: After you eat all the apples, the following
| statements are true: - there are no apples left - there are 0
| apples - there are 0 oranges
|
| The following statement does not apply though: - there are no
| oranges left ... as this implies that there have been oranges
| available at some time before.
| amelius wrote:
| But that last statement has more to do with the word "left"
| than with "zero".
| pflenker wrote:
| True - the point I am trying to underscore is that the
| concept of 0 is not the same as counting down and
| arriving at ,,nothing". These are two related, but not
| equal, concepts.
| adrian_b wrote:
| In both of your examples, 0 is the result of a counting
| operation. Only what is counted differs.
|
| "There are 0 apples left." is the answer to the question
| "How many apples are left?".
|
| "There are 0 oranges." is the answer to the question "How
| many oranges are on the table?" (or "in the box" or
| wherever).
|
| Everywhere where 0 appears in speech, it is the result of a
| counting or measuring operation, which provides the answer
| to a question, expressed or implied.
|
| That counting or measuring operation could have had any
| other number as its result, instead of 0, which
| demonstrates that the nature of 0 is the same as that of
| any other cardinal number, i.e. it is a quantity (term
| introduced already by Aristotle, in his "Categories", where
| the various kinds of concepts and the words that name them
| were classified by the kinds of questions to which they
| provide answers).
| pflenker wrote:
| That is true, but missing the point made by the parent
| comment. Being familiar with 0, we can clearly see how it
| follows from counting down. We fail to see why this would
| be ever difficult, since we're so used to this concept.
|
| However, this does not match the reality of how we learn
| about 0. Children learn 0 conceptually way after learning
| to count, because the concept is trickier to grasp.
| adrian_b wrote:
| What you say about children is true and I have already
| said in my first comment that the structure of the words
| meaning "zero" in most old recorded languages indicates
| that the concept of "zero" must be more recent than the
| concepts of the other small numbers and of negation.
|
| Nevertheless, even if the concept of zero has been
| understood later than the concepts of 1, 2, 3 ..., it has
| still been understood in many places at least four
| thousand to five thousand years ago, i.e. already by the
| time of the oldest writings that have been preserved and
| thousands of years before the invention of the positional
| system of writing numbers, where the importance of zero
| has greatly increased and where it has required a
| dedicated graphic symbol.
| Asraelite wrote:
| When there is one (or more) of something, you can observe the
| thing to say what there is one of. I see an apple, I can say
| "one apple".
|
| When there is zero of something, what is there zero of? That
| question becomes a lot more difficult to answer. In theory you
| could say there is zero of anything not observed, but this
| isn't a very precise or useful definition and isn't what's done
| in practice. The things we choose to actively describe there
| being zero of depends on knowledge of what existed previously,
| on cultural and linguistics norms, on context etc.
|
| For example, I can say "there are zero apples in the basket",
| but this requires me to know that there previously existed
| apples in the basket as opposed to oranges, or that the
| addressee of my statement was expecting apples to be there.
| This knowledge wouldn't be required if there was at least one
| apple.
|
| Using zero fundamentally requires more mental reasoning than
| using small positive integers.
| loa_in_ wrote:
| Level 1:
|
| There's something or there's nothing
|
| Level 2 if there's something:
|
| There's one or more of this thing and it's not going to
| transform into same amount of something different.
| ForOldHack wrote:
| You seem to have escaped cultural zeros. In Hitchhikers
| guide... You must prove to Marvin, that you are
| intelligent, by having tea and no tea, at the same time. A
| paradox and a cultural reference .
|
| Also with stolen paintings: you have no painting,but you
| have evidence of it's existence.
|
| And lastly, in song: "can't buy me love."
|
| And of course physics: T(sub) 0. As well as space flight:
| t-zero.
| BobaFloutist wrote:
| If I have one apple I still have zero oranges.
| crazygringo wrote:
| > _When there is zero of something, what is there zero of?
| That question becomes a lot more difficult to answer._
|
| > _Using zero fundamentally requires more mental reasoning
| than using small positive integers._
|
| I disagree. Positive integers don't exist in a vacuum either.
|
| You always start with a context -- the thing you're counting
| -- and then you are given a number. Whether 3 or 0.
|
| I see no greater conceptual complexity in either case. I
| think the flaw is in your sentence:
|
| > _This knowledge wouldn 't be required if there was at least
| one apple._
|
| Yes it is required. We're not comparing the statements "0
| apples" and "1", we're comparing "0 apples" and "1 apple".
| Asraelite wrote:
| I figured somebody might try to make this argument.
|
| The problem with your comparison is that you're already
| starting from too much.
|
| > You always start with a context
|
| And where does that context come from? The true starting
| point is nothing. The context needs to arise from
| something. In order to even form a thought about apples and
| not any of the other thousands of concepts that you're
| aware of in the first place, you need a prompt.
|
| In the case of an existing object, the prompt is seeing it.
| In the case of a non-existing object, the prompt is the
| combination of social and memory factors that I mentioned
| before. The former is simple, the latter is complex.
|
| Another thing to consider: humans invented the number zero
| long after the natural numbers. Children learn to
| understand it after the natural numbers. The brain
| processes it differently, as mentioned by the article.
| There is overwhelming evidence that yes, zero is more
| complicated than the natural numbers for humans to think
| about.
|
| If your reasoning ends with the conclusion that it's not
| actually more complicated, then that is in direct conflict
| with the evidence and it shows that you must be missing
| something.
| crazygringo wrote:
| > _In the case of an existing object, the prompt is
| seeing it._
|
| Not necessarily, not at all. It might be, "hmm, do I have
| any apples left in the sack?" Or "did my daughter
| retrieve the ears of corn"? Sometimes we randomly come
| across an object, but in a great many cases (the
| majority?) we already have the context -- we know what we
| expect to see (or not see), or what we're looking for, or
| what we're investigating. Humans are goal-oriented
| creatures; we're not just responding to current sensory
| stimuli.
|
| > _humans invented the number zero long after the natural
| numbers_
|
| Only in the highly technical sense of a dedicated symbol
| for balanced financial accounts, or a digit placeholder
| in a positional number system. Languages all have
| everyday linguistic equivalents for zero like "none",
| "no", "aren't any", etc. These mean "zero" in the
| counting sense, precisely and exactly. There's no
| evidence at all that these came after something like the
| number 7.
|
| > _and it shows that you must be missing something_
|
| Or you yourself, in this case. You are unfortunately
| looking at an overly limiting definition of context, an
| overly limiting definition of zero, and you're missing
| important parts of linguistic history.
| paganel wrote:
| > There are zero apples.
|
| "are" and "zero", meaning non-existence/non-"are"-ness, in the
| same sentence is seen as strange by many, yes, and that include
| me.
| cwmoore wrote:
| You can have zero of everything in the same place and time.
| paganel wrote:
| Which helps make my point, i.e. your Parmenidian view on
| this problem.
| Drakim wrote:
| Five apples is not the same as five oranges, but zero apples is
| the same as zero oranges. After all, what would be difference
| between a whole bunch of nothing and another whole bunch of
| nothing? Zero is typeless, which makes it unique and not just
| another quantity.
| BobaFloutist wrote:
| Zero oranges is different from zero apples, because you can
| have ten apples and zero oranges.
| thfuran wrote:
| I have ten apples sitting right next to zero apples.
| kylehotchkiss wrote:
| There are zero horses in my room. The absence of something is a
| default condition.
| bmacho wrote:
| There is no strangeness of zero. It is very natural. You use it
| to count objects. 0,1,2, .. they have a good physical model:
| number of stones. It's just the teachers that insist on confusing
| kids. Also they are commonly called as 'natural numbers'.
| (Acknowledging the fact, that how they are called can't be a
| basis of an argument.)
|
| Negative numbers however. They have no good physical models, at
| least that I know of. They are mostly just a tool to make
| accounting easier, or to denote one out of 2, indistinguishable
| cases (direction on the axis x, direction of angular velocity,
| electric charge). But the teachers don't insist confusing the
| kids with them.
| saurik wrote:
| (FWIW, the "natural numbers" in fact does not include 0: if you
| include 0 you get the "whole numbers". I otherwise agree with
| you that 0 isn't at all strange.)
| 4ad wrote:
| The definition of natural numbers used in any sort of
| formalized mathematics includes zero. Without zero, you don't
| have induction (and N wouldn't form a monoid).
| scapp wrote:
| Nothing wrong with starting at 1 for induction, but yes,
| having an additive monoid is nice (still get a
| multiplicative monoid with N*)
| bmacho wrote:
| I did not know that "whole numbers" refer to nonnegative
| integers too.
|
| > The whole numbers were synonymous with the integers up
| until the early 1950s. In the late 1950s, as part of the New
| Math movement, American elementary school teachers began
| teaching that whole numbers referred to the natural numbers,
| excluding negative numbers, while integer included the
| negative numbers. The whole numbers remain ambiguous to the
| present day.
|
| https://en.wikipedia.org/wiki/Integer
|
| > In mathematics, the natural numbers are the numbers 0, 1,
| 2, 3, and so on, possibly excluding 0. Some start counting
| with 0, defining the natural numbers as the non-negative
| integers 0, 1, 2, 3, ..., while others start with 1, defining
| them as the positive integers 1, 2, 3, ... . Some authors
| acknowledge both definitions whenever convenient. Sometimes,
| the whole numbers are the natural numbers plus zero. In other
| cases, the whole numbers refer to all of the integers,
| including negative integers. The counting numbers are another
| term for the natural numbers, particularly in primary school
| education, and are ambiguous as well although typically start
| at 1.
|
| https://en.wikipedia.org/wiki/Natural_number
|
| What a shame.
| messe wrote:
| It's dependent on your choice of definition. Most include it.
| 8bitsrule wrote:
| When I go to make a cake and need another cup of sugar, I have
| zero sugar. When I borrow a cup from the neighbor, and owe them
| one, I have -1 cups of sugar. Naturally.
| 4ad wrote:
| Yeah, in many ways the concept of a group is far more natural
| to us than the concept of natural numbers. I don't know why
| people make such a big deal of this, and why it's so badly
| explained in school.
| bmacho wrote:
| You can give your kid candies. What is the equivalent of
| this with groups?
| 4ad wrote:
| Playing with trains on a track.
| ForOldHack wrote:
| Brilliantly.
|
| This sentence no verb. Self referentially.
| hgomersall wrote:
| You get a long way if you start from geometry, the numbers just
| being written representation of geometric constructs. The
| numberline is one such geometric construct that can be
| interpreted easily with the reals (distance along the line from
| the origin in some direction). Complex numbers can then be
| considered an algebra representing rotations and translations
| in a plane. The various geometric algebras map to more
| geometric constructs again.
| pwdiscflatmajor wrote:
| I've heard it said that English tends to reify language; that it
| tends to turn every-thing into a "thing". So no-thing still gets
| processed as a thing, in English. I'd be curious how "nothing"
| and "zero" are processed in other languages/cultures and if the
| experience/phenomenology changes. E.g., if 0 is recorded state of
| a place-value on an abacus, that's got a very different feel than
| 0 as a count term. And the 0th position in an array has a very
| distinct feeling as a location, and say the transition from an
| empty basket to a basket with one apple, has it's own *feel.
| Emptiness is a state of being, not a "thing". Not every-thing is
| a thing, but English makes it seem so. Maybe all languages make
| it so seem (And maybe German too, [c.f., Heidegger: the nothing
| noths])?
|
| *edited to say feel instead of fill
| pwdiscflatmajor wrote:
| ...and treating a term for a state of being as if the state we
| an object, a thing, vs the state a thing is in, is a category
| mistake (and perhaps one that only happens with
| overthinking)--like what I might be doing right now? is what
| I'm _doing_ right now a "thing"? No, right? Because, verbs
| aren't nouns. Are the states of things things? Are adjectives
| nouns? Are verbs nouns?
|
| (no, but since in English I can verb my adjectived nouns into
| all sorts of noun-y verb craziness, nothing can get nouned and
| verbed out of adjectiving all day long, because English.)
| red75prime wrote:
| I'm not a fan of Sapir-Whorf hypothesis. People extend the
| language to fit their needs, not other way around. "Turning
| everything into a thing" is probably the inevitable way to
| accommodate philosophical thinking. Not everything is a thing,
| but to be able to make anything into an object of consideration
| and communication you need a word for it and a word is a thing.
| sph wrote:
| Zero is pretty easy to understand if you add a layer of
| indirection: nothingness is hard to imagine, but add a container
| around it, and that's just an empty container. Zero is the empty
| glass.
| ForOldHack wrote:
| Zero is also a missing painting in the Louvre. The spot on the
| wall where the Mona Lisa hung. People would state at the spot
| as if it was there.
|
| I wonder where zero is when I point a web cam at a screen and
| see the tunnel effect. Am I, the observer the zero camera?
| ForOldHack wrote:
| Further: the day the war ended was day 0, then came the first
| day after the war. Now for historical context, on which day
| did the Mona Lisa return?
| adrian_b wrote:
| There was nothing strange about zero for our distant ancestors.
|
| There are only two things that have been understood much later
| about zero, that it is a quantity that in many circumstances
| should be treated in the same way as any other numbers and that
| it requires a special symbol in order to implement a positional
| system for writing numbers.
|
| The concept of zero itself had already been known many thousands
| of years before the invention of the positional system for
| writing numbers.
|
| All the recorded ancient languages were able to answer to the
| question "How many sheep do you see there?" not only with words
| meaning "one", "two", "three", "four", "few", "many" and so on,
| but also with a word meaning "none". Similarly for the answers to
| a question like "How much barley do you have?".
|
| Nevertheless, the concept of the quantity "zero" must have been
| understood later than the concepts of "one", "two", "many" and of
| negation, because in most languages the words similar in meaning
| to "none", "nobody", "nothing", "null" are derived from negation
| words together with words meaning "one" or denoting small or
| indefinite quantities.
|
| Because the first few numbers, especially 0, 1 and 2 have many
| distinctive properties in comparison with bigger numbers, not
| only 0 was initially perceived as being in a class somewhat
| separate from other numbers, but usually also 1 and 2 and
| sometimes also 3 or even 4.
|
| In many old languages the grammatical behavior of the first few
| numbers can be different between themselves and also quite
| different from that of the bigger numbers, which behave more or
| less in the same way, consistent with the expectation that the
| big numbers have been added later to the language and at a time
| when they were perceived as a uniform category.
| bugbuddy wrote:
| There are other factual problems with this article. Most
| jarring is that it repeats this old incorrect information:
|
| > Inside the Chaturbhuj Temple in India (left), a wall
| inscription features the oldest known instance of the digit
| zero, dated to 876 CE (right). It is part of the number 270.
|
| This is the real oldest known zero:
|
| > Because we know that the Caka dynasty began in AD 78, we can
| date the artifact exactly: to the year AD 683. This makes the
| "0" in "605" the oldest zero ever found of our base-10, "Hindu-
| Arabic" number system.
|
| > But the Cambodian stone inscription bears the first known
| zero within the system that evolved into the numbers we use
| today.
|
| Maybe the Khmers came up with it or maybe they got the idea
| from India. Either way, we should set the facts straight.
|
| https://time.com/3845786/my-quest-to-find-the-first-zero/
| westurner wrote:
| 0 (number) > Classical antiquity;
| https://en.wikipedia.org/wiki/0#Classical_antiquity :
|
| > _By AD 150, Ptolemy, influenced by Hipparchus and the
| Babylonians, was using a symbol for zero ( -- deg ) [25][26]
| in his work on mathematical astronomy called the Syntaxis
| Mathematica, also known as the Almagest. [27] This
| Hellenistic zero was perhaps the earliest documented use of a
| numeral representing zero in the Old World. [28]_
| poincaredisk wrote:
| >In many old languages the grammatical behavior of the first
| few numbers can be different between themselves
|
| Not even old languages, in many languages today (notably
| slavic) nouns decline differently with "1", "2", "3", "4" and
| say "5".
|
| In many languages, including existing ones (Lithuanian, Irish
| and Slovenian come to mind) there exists a concept of a
| grammatical number "dual", in addition to singular and plural.
| david_allison wrote:
| For anyone looking for more information:
|
| https://www.unicode.org/cldr/charts/46/supplemental/language.
| ..
|
| grep for "two" (note: "two" means 'dual', not literally '2')
|
| Spec: https://cldr.unicode.org/index/cldr-spec/plural-rules
| cperciva wrote:
| _Not even old languages, in many languages today (notably
| slavic) nouns decline differently with "1", "2", "3", "4" and
| say "5"._
|
| Hence a wonderful piece of Soviet humour: A factory needs 5
| fireplace pokers, but none of the workers knows the correct
| plural form for 5 fireplace pokers; not wanting to appear
| ignorant when they send their request to management, they
| request "3 fireplace pokers and 2 more". Some months later
| they receive the fireplace pokers with a note saying "here
| are 4 fireplace pokers and 1 more" -- because management
| didn't know the word either!
| TheRealPomax wrote:
| Except there is, because the zero the article is talking about
| is the mathematical zero, not the natural language one. In your
| experience of natural language, zero is a quantity. In math,
| it's _very much_ not: it 's a singularity.
|
| Sure, as "that number between -1 and 1 on the number line",
| zero is mostly unremarkable, but math doesn't really care about
| number lines that much: as a singularity that arises from
| mathematical operations, its behaviour, and its "so what _is_
| it, really? " is a proper brain twister.
| drewcoo wrote:
| Zero was not a "counting number." It wasn't used in early
| calculations.
|
| To an ancient Roman, tomorrow is 2 (ok, II) days from now.
| Today . . . one . . . tomorrow . . . two . . . two days!
|
| > Because the first few numbers, especially 0, 1 and 2
|
| Considering zero one of "the first few numbers," or even
| considering zero being a number, or for that matter considering
| zero at all are very recent concepts for humans.
|
| A millennium ago, asking someone to count all the sheep in a
| field that had no sheep would be like today asking the sound of
| one hand clapping.
| johnisgood wrote:
| I have a hard time believing this though. If there are no
| sheep, then would it really not be "obvious" to them that
| there are no sheep, i.e. zero sheep, or none to be found?
| What would have been their answer if not something akin to
| this?
| thfuran wrote:
| It's not that there's no concept of the absence of sheep,
| it's that there's no concept that that can be represented
| as some number of sheep.
| adrian_b wrote:
| This may sound clever, but it fails the experimental
| tests.
|
| In most modern grammar theories and already in some of
| the works of the Ancient Greeks and Romans, a group of
| words is recognized as having the same nature if any of
| those words can be substituted for another of them in a
| given sentence.
|
| Given the sentences (which have one-to-one
| correspondences in the ancient languages):
|
| I see zero sheep. (In Latin, "null-" is the word stem for
| zero.) I see one sheep. I see two sheep. I see three
| sheep. ...
|
| both a modern grammarian and also some of the ancient
| ones will conclude that zero, one, two, three ... have
| the same nature.
|
| So zero was a concept that represented a certain quantity
| of sheep.
|
| The Romans might have not used the word "number" in
| reference to zero, but there is no doubt that they
| understood the concept of a set whose members are 0, 1,
| 2, 3 and so on.
|
| The proof is that the normal way of saying "some" in
| Latin, i.e. an unknown number that is at least 1, was to
| say "not zero", i.e. by identifying the subset 1, 2, 3,
| ... with the complement of the subset that contains 0 in
| the set of what now are called natural numbers.
|
| While they did not have standardized words for concepts
| like set, subset, complement, membership or quantifiers,
| these concepts of the modern set theory were nevertheless
| well understood by the educated people in the Ancient
| World as they were studied within Logic a.k.a. Dialectic.
| adrian_b wrote:
| A millennium ago or five millennia ago it would have been
| very normal to ask someone to count all the sheep in a field
| that had no sheep, because the one asking would not have
| known that there were none, hence the question. The asked
| would also have not been surprised by the question, because
| it would have been obvious that whoever had asked had not
| known the answer, and they would have provided the correct
| answer: none.
|
| Already before the ancient Romans, Aristotle had made a
| classification of the parts of speech based on their meaning
| as exposed by the kind of questions to which they can provide
| answers (the so-called "Categories" of Aristotle). For some
| of his categories he has coined new names, including the name
| that has been later translated into Latin as "quantity", and
| Aristotle has divided the quantities into discrete quantities
| (providing answers to "How many?") and continuous quantities
| (providing answers to "How much?"). According to Aristotle's
| definitions, "zero" was certainly a quantity, together with
| the other numbers and with other words that can express
| quantities.
|
| I do not think that there is any preserved text from the
| Classical period of Ancient Greece in which it is stated
| whether they thought that zero belongs to what they called
| numbers (arithmoi), or not. In any case whether they had
| chosen to call zero using the term "number" or not is less
| important than the fact that they understood perfectly the
| more general concept of "quantity", which included various
| subclasses, like numbers (what we call integer numbers) and
| measures (what we call real numbers), and to which zero also
| belonged.
|
| About the ancient Romans, the most frequent way of saying in
| Latin "some", i.e. an unknown number, but at least 1, was
| "not zero". Therefore regardless whether they were using the
| word "number" for zero or not, they had a clear concept of a
| set consisting of 0, 1, 2, 3 and all the other cardinal
| numbers, so they identified the subset with 1, 2, 3 ... as
| the complement of the subset containing zero.
|
| The ancient people were much less ignorant than some modern
| people believe.
| jaggajasoos33 wrote:
| In the context of brain zero is much older than 7th century.
| Buddhist philosophy revolves around "Shunyaataa" (Zeroness -
| Shunya means zero in Sanskrit). Shunya itself comes from the root
| Svi - empty/hollow.
|
| In the context of brain, Buddhism and various branches of
| Hinduism spent a lot of time pondering. Several meditative
| techniques involve meditating over this hollowness or the
| ultimate absence.
|
| I do know that Zero the numerical placeholder is different from
| Buddhist idea of "Zeroness" but in the context of brain it should
| be similar I feel. Absence of something means cardinality of zero
| but Buddhists literally asked this question about reality itself.
| hbarka wrote:
| Strangeness of zero or strangeness of null?
| erie wrote:
| It is strange that when Arabic and Hebrew languages are written
| from left to write, that is not the case with numbers, a common
| saying in Arabic is 'zero on the left', that means worthless.
| Arabs in old times hated zero because it signified no gains in
| trade and wars, and was called a vacuum sign because any number
| multiplied by it ended up with nothing, the Arabic zero word
| 'cepher ' was a translation of Sanskrit word sunya that meant
| empty. The first English use of zero was in 1598. Al-Khwarizmi or
| Algorismus made common in use in 773 in Baghdad.
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