Exploring the Existence of Zero

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In the another thread, I queried some posters comments which were along the lines that zero is a metaphysical concept / doesn't exist.

Baywax responded, as below, and my additional comments are in blue.

Zero exists as much as the number 1 in that these are both language equivalents that correspond to a quantity of actual "things". So, in the sense that "zero" exists as a stimulated group of neurons... beyond that it is only a description of a quantity or a lack of quantity.

I presume you mean a stimulated group of neurons in the brain .. but then, everything exists as that. 'Cat' and 'weather' and 'syndrome' also exist as a stimulated group of neurons in the brain. So how is this unique to zero ?

In reality the description "zero" describes the "non-existence" of quantity and so even the "thing" or "quantity" that zero describes... does not exist.

Zero describes non existence. OK.

There is one apple on the table. I remove it. There is zero apples on the table. The 'zero' character of the apple (in realtion to the table) is now very real.What's wrong with that ? What am I missing ? Is the issue here being complicated more than it should ?

 

[Galap]

Zero is a number like any other. All numbers are abstract concepts which can be applied to give them physical meaning. Natural numbers have easy physical meanings, like there are 2 apples on the table. The number means, well, the number of them. Rational numbers can be used to measure things, like the bag of apples weighs 1.2432 Newtons.

Even imaginaries have physical meaning with things like radio signals and electricity.

The point I'm trying to make is that all numbers have different physical meanings for different things. Zero is no different, only the meaning is a little different.

 

[pallidin]

Right. Zero is a little special.
Whereas you can have 1 apple and easily add more, you are maintaining the concept of an apple throughout, as at least one apple always exists in current observation.

When going from one or more apples to zero, it gets slightly more complicated, as the current observation of zero apples does not directly imply there were apples before. It is inferred from previous observation or "acceptance" if there were no observation.

 

[ThomasEdison]

The Book of Nothing by John Barrow delved into this subject quite a bit.
https://www.amazon.com/dp/0375420991/?tag=pfamazon01-20

It was the first physics (and philosophy of science) book for laymen that I read that got me interested in science again.

What is interesting are which cultures accepted zero and which did not.

The Aztecs had it.

 

[alt]

The Book of Nothing by John Barrow delved into this subject quite a bit.
https://www.amazon.com/dp/0375420991/?tag=pfamazon01-20

It was the first physics (and philosophy of science) book for laymen that I read that got me interested in science again.

What is interesting are which cultures accepted zero and which did not.

The Aztecs had it.

Thanks. Which cultures didn't have it ? Also, are there any that presently don't ?

 

[alt]

Right. Zero is a little special.
Whereas you can have 1 apple and easily add more, you are maintaining the concept of an apple throughout, as at least one apple always exists in current observation.

When going from one or more apples to zero, it gets slightly more complicated, as the current observation of zero apples does not directly imply there were apples before. It is inferred from previous observation or "acceptance" if there were no observation.

Thanks. That's what I like about this place - it forces you (well, me) to exersise the grey matter a little more than usual.

In fact, zero apples on the table does have to assume that there were some there in the first place to make sense .. I think ..

 

[nismaratwork]

http://yaleglobal.yale.edu/about/zero.jsp

The Sumerians were the first to develop a counting system to keep an account of their stock of goods - cattle, horses, and donkeys, for example. The Sumerian system was positional; that is, the placement of a particular symbol relative to others denoted its value. The Sumerian system was handed down to the Akkadians around 2500 BC and then to the Babylonians in 2000 BC. It was the Babylonians who first conceived of a mark to signify that a number was absent from a column; just as 0 in 1025 signifies that there are no hundreds in that number. Although zero's Babylonian ancestor was a good start, it would still be centuries before the symbol as we know it appeared.

The renowned mathematicians among the Ancient Greeks, who learned the fundamentals of their math from the Egyptians, did not have a name for zero, nor did their system feature a placeholder as did the Babylonian. They may have pondered it, but there is no conclusive evidence to say the symbol even existed in their language. It was the Indians who began to understand zero both as a symbol and as an idea.

Brahmagupta, around 650 AD, was the first to formalize arithmetic operations using zero. He used dots underneath numbers to indicate a zero. These dots were alternately referred to as 'sunya', which means empty, or 'kha', which means place. Brahmagupta wrote standard rules for reaching zero through addition and subtraction as well as the results of operations with zero. The only error in his rules was division by zero, which would have to wait for Isaac Newton and G.W. Leibniz to tackle.

But it would still be a few centuries before zero reached Europe. First, the great Arabian voyagers would bring the texts of Brahmagupta and his colleagues back from India along with spices and other exotic items. Zero reached Baghdad by 773 AD and would be developed in the Middle East by Arabian mathematicians who would base their numbers on the Indian system. In the ninth century, Mohammed ibn-Musa al-Khowarizmi was the first to work on equations that equaled zero, or algebra as it has come to be known. He also developed quick methods for multiplying and dividing numbers known as algorithms (a corruption of his name). Al-Khowarizmi called zero 'sifr', from which our cipher is derived. By 879 AD, zero was written almost as we now know it, an oval - but in this case smaller than the other numbers. And thanks to the conquest of Spain by the Moors, zero finally reached Europe; by the middle of the twelfth century, translations of Al-Khowarizmi's work had weaved their way to England.

http://math.suite101.com/article.cfm/the_number_zero

The Ancient History of Zero
In reality, the number zero hadn’t been accepted into any known mathematical system of numerals until sometime in the 9th century A.D., and the best evidence is that it first became fully recognized as a numeral in India. A few centuries prior to this time the Romans had found a way to recognize nothing in their Roman Numeral system, but instead of receiving its own symbol, it was denoted by a word – nulla (meaning “nothing”). Prior to this time, different cultures with different mathematical traditions had found any number of different “tricks” to enable them to represent the absence of numbers, though none had accepted zero as a number in itself.

For some of the history, and its late emergence as a discrete concept.

For fun: http://home.ubalt.edu/ntsbarsh/zero/zero.htm

 

[jreelawg]

You could practically define 0, as "doesn't exist" couldn't you. For example, 0 apples means apples don't exist. Zero apples in the basket means apples don't exist in the basket.

So O itself, exists as a symbol, used to describe the non-existence of something.

 

[nismaratwork]

You could practically define 0, as "doesn't exist" couldn't you. For example, 0 apples means apples don't exist. Zero apples in the basket means apples don't exist in the basket.

So O itself, exists as a symbol, used to describe the non-existence of something.

As I pointed out in another thread, I find the concept of nothingness as unfathomable as something. "Doesn't Exist", or rather "no quantity" is a powerful concept, because how do you define an absence of apples without a reference to apples? Nothing is a foreign concept to us, and probably anything that exists. This ignores the mathematical issues with 0, which are many and sundry.

 

[jreelawg]

Reminds me of a thread, about whether nothing is something or not.

 

[nismaratwork]

Reminds me of a thread, about whether nothing is something or not.

My kneejerk reaction is that nothing is nothing, but I think in the end that debate can only ever be semantic; we lack the capacity to envision or articulate nothingness with words; only mathematical symbols give us that ability to some degree.

 

[alt]

http://yaleglobal.yale.edu/about/zero.jsp




http://math.suite101.com/article.cfm/the_number_zero



For some of the history, and its late emergence as a discrete concept.

For fun: http://home.ubalt.edu/ntsbarsh/zero/zero.htm

What I think I'm getting is that in most cases, 'zero' does not denote complete and absolute absence - perhaps because others in other times, could not imagine such a thing either.

Speaking fluent Greek, I am often interested in how closely our English words and the ideas behind them, correspond to the ideas behind the same word but in another, perhaps older language. It is interesting that often, those ideas are not the same, as one would expect that they should be.

I thought it worthwhile to check out 'zero' in the Greek. Firstly, without looking anything up, I immediately know that the common translation for 'zero' is 'miden'. So, I go to ..

http://www.kypros.org/cgi-bin/lexicon

and I get;

zero - μηδέν, μηδενικό (miden, mideniko)

.. which strikes me, as pretty darn close to our 'median' (though ain't it funny that although I've used that word in the Greek often, I've never previously made that connection). So no absolute absence here, and in fact, one could even say 'a neutral / median point' - zero being the median of all numbers 'n all !

I tried to find something a little earlier (more ancient), at ..

http://www.lexilogos.com/english/greek_ancient_dictionary.htm

.. and got;

zero - οὐδείς (no transliteration given in that dictionary, but I offer "oudes")

I can't think of any English words with as immediate a correspondent sound (as in 'miden / median'), so still searching for the .. um .. flavour of the word 'οὐδείς', I think of Greek words that might contain it. And the most obvious to me is;

'οὐδέτερος' - (oudeteros) not either, neither of the two

.. and today, used mainly to indicate neutrality. If you abstained from voting at a meeting for instance, you are 'οὐδέτερος'. In fact, it is also specifically used to denote gender; arseniko - male, thilyko - female, oudetero - neutral (Greek language is very heavy in it's use of gender).

Could be interesting too, that 'hades' - the world of the dead, is probably related to 'οὐδείς' though I haven't perused this.

So anyhow, what I'm saying, is than neither 'μηδέν' nor 'οὐδείς' have a flavour of absolute nothingness about them, and in fact, seem to range around neutrality, non involvement, middle points .. stop me if I'm getting ahead of myself - please !

I just find this stuff quite fascinating. Zero zero !

 

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You could practically define 0, as "doesn't exist" couldn't you. For example, 0 apples means apples don't exist. Zero apples in the basket means apples don't exist in the basket.

So O itself, exists as a symbol, used to describe the non-existence of something.

describe the non-existence of something

Therein lies the conundrum.

 

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As I pointed out in another thread, I find the concept of nothingness as unfathomable as something. "Doesn't Exist", or rather "no quantity" is a powerful concept, because how do you define an absence of apples without a reference to apples? Nothing is a foreign concept to us, and probably anything that exists. This ignores the mathematical issues with 0, which are many and sundry.

This ignores the mathematical issues with 0, which are many and sundry

Some simple ones please (befitting the mathematical abilities of an orangutang) ?

 

[luma]

this ignores the mathematical issues with 0, which are many and sundry

some simple ones please (befitting the mathematical abilities of an orangutang) ?

0/0 .

 

[nismaratwork]

0/0 .

Or n/0 for which some number multiplied by zero is not 0. Both work!

 

[alt]

Or n/0 for which some number multiplied by zero is not 0. Both work!

0/0 = 0 ? I can't see the issue here.

I can see the issue with n/0 though - that is, presuming I'm on the right track, that being ..

6 / 3 = 2, therefore, 2x3 must equal 6. But,

6 / 0 = 6, therefore 0x6 must equal 6 .. NOT !

Is that the oddity ?

PS - you said in another thread as I recall, that you liked wordplay.

I would be interested to know your view on my post a few up, where I pointed out that in the Greek, zero is 'miden' which I feel sure, relates to our 'median' - thus, tending towards something much different from absolute absence (IMO). Whats your opinion ?

 

[nismaratwork]

0/0 = 0 ? I can't see the issue here.

I can see the issue with n/0 though - that is, presuming I'm on the right track, that being ..

6 / 3 = 2, therefore, 2x3 must equal 6. But,

6 / 0 = 6, therefore 0x6 must equal 6 .. NOT !

Is that the oddity ?

PS - you said in another thread as I recall, that you liked wordplay.

I would be interested to know your view on my post a few up, where I pointed out that in the Greek, zero is 'miden' which I feel sure, relates to our 'median' - thus, tending towards something much different from absolute absence (IMO). Whats your opinion ?

I do enjoy wordplay, and my view of your analysis is that you're probably correct. If you consider the use of the abacus or antikythera as a device for calculation, an even distribution would be a null result. This is just based on what you've said, but it makes sense, and fits with historical records that the concept of zero wasn't around in ancient Greece.

your "6/0 therefore 0x6" is the conundrum I was pointing out, but also, what is nothing divided by nothing? More nothing, less nothing, the same quantity of no quantity? Mathematically it isn't significant the way that an integer/0 is, but it raises the more philosophical end of the spectrum. I'm very glad you started this thread.

 

[zomgwtf]

woah
n/0 = n? I must've missed that in all my math courses.

Division by 0 isn't a legitimate mathematical process as far as I know.
0/0 does not = 0

The only way you can divide by 0 is by using special mathematical theories where division is slightly modified.

 

[BruceG]

Taking a wider view of the role of zero in physics (beyond counting apples) reveals it plays an extremely constructive role. Consider how zero comes into the following statements taken at random from a textbook on string theory:

1. Action is a dimensionless quantity

2. The mass-shell condition is p^2 + m^2 = 0

3. The classical string motion extremizes the world-sheet area (dS = 0)

4. Consistency of the p-brane action requires the cosmological constant to vanish when p=1.

5. The ground state |0> satisfies a(m)|0> = 0 for m > 0.

etc, etc..must I go on.

 

[disregardthat]

0 is a symbol. Often in association with a formal set of operational rules. What it represent is not a mathematical problem. It may represent whatever you want, accurately or not. There are no mathematical problems with 0, it behaves just like we want.

 

[apeiron]

My kneejerk reaction is that nothing is nothing, but I think in the end that debate can only ever be semantic; we lack the capacity to envision or articulate nothingness with words; only mathematical symbols give us that ability to some degree.

Treating the question as a philosophical rather than a mathematical one, the "least" state of things is a vagueness.

As people realize, to have a local state of nothing (a missing apple) we need a global state, a general context, in which this missing apple can be noticed. The absence has to be counted as a definite event. We are saying, there are none of these things.

So being able to count nothing is a crisp action (I looked within a context and found it bare of events that could have logically been there).

To really get rid of thing-ness, we have to erase the context as well as its possible events. And this is why a vagueness is even less than nothing. We are now asking the question, is there even a context there which could contain events, and so not contain them? Well maybe, maybe not. It is just looking vague - a raw state of what might be potential.

 

[disregardthat]

Nothingness in language is always used in a specific context. I can hardly see or understand any usage of a general sense of 'nothingness'. In referrance to any group of abstract objects, nothing is used as the absence of such.

 

[pallidin]

Be reminded that "nothingness" is a comparative concept.
That is, we always talk about "nothingness" with respect to "something"

 

[apeiron]

I can hardly see or understand any usage of a general sense of 'nothingness'.

That is the point. First we note that the standard way of thinking about nothing must depend on a context - which is then some kind of something. So either you stop there (cease to philosophise) or you wonder about how even the context can be erased, devolved, somehow melted away, too.

You are asking how can there be an absence of even absence? Not just a localised absence but a global one.

It would seem clear that there can't be. Which is where turning in a new direction becomes fruitful. If instead of thinking in terms of things that either exist or don't exist, you take the developmental view in which things are first merely possible, then later become actual.

So at first, anything and everything was possible (including the local existence of absences). Then later, this potential became translated into an actual state of affairs.

Vagueness thus becomes your primal potential, how things "are" before either existence or non-existence can crisply apply as descriptions. It is the most primitive level of "being" we can imagine, and certainly more primitive than nothing-ness (which requires the existence of a global context to itself exist).

 

[wrongusername]

The only way you can divide by 0 is by using special mathematical theories where division is slightly modified.

What are some examples of such math theories?

 

[disregardthat]

What are some examples of such math theories?

http://en.wikipedia.org/wiki/Riemann_sphere

 

[alt]

I do enjoy wordplay, and my view of your analysis is that you're probably correct. If you consider the use of the abacus or antikythera as a device for calculation, an even distribution would be a null result. This is just based on what you've said, but it makes sense, and fits with historical records that the concept of zero wasn't around in ancient Greece.

Do you think that the ancients had not even a concept of absolute nothingness ? Or are you saying their 'even distribution / nul result' was merely for practical purposes ?

your "6/0 therefore 0x6" is the conundrum I was pointing out, but also, what is nothing divided by nothing? More nothing, less nothing, the same quantity of no quantity? Mathematically it isn't significant the way that an integer/0 is, but it raises the more philosophical end of the spectrum. I'm very glad you started this thread.

Yes, the 'nothing' conundrum.

Nothing is better than complete happiness in life.
A ham sandwich is better than nothing.
Therefore, a ham sandwich is better than complete happiness in life.

Where do we go from here ?

 

[alt]

That is the point. First we note that the standard way of thinking about nothing must depend on a context - which is then some kind of something. So either you stop there (cease to philosophise) or you wonder about how even the context can be erased, devolved, somehow melted away, too.

You are asking how can there be an absence of even absence? Not just a localised absence but a global one.

It would seem clear that there can't be. Which is where turning in a new direction becomes fruitful. If instead of thinking in terms of things that either exist or don't exist, you take the developmental view in which things are first merely possible, then later become actual.

So at first, anything and everything was possible (including the local existence of absences). Then later, this potential became translated into an actual state of affairs.

Vagueness thus becomes your primal potential, how things "are" before either existence or non-existence can crisply apply as descriptions. It is the most primitive level of "being" we can imagine, and certainly more primitive than nothing-ness (which requires the existence of a global context to itself exist).

Your post above, and your post #22 are fascinating - certainly turning in a new direction, as you say. I'm struggling to uderstand one or two of your ideas, and will continue to re-read until I hopefully do understand them more.

PS - your last two paragraphs above, reminded me of the 1st chapter, 1st few verses in Genesis for some reason (I think I read elsewhere that you're atheist- so no offence is intended here )

 

[nismaratwork]

Do you think that the ancients had not even a concept of absolute nothingness ? Or are you saying their 'even distribution / nul result' was merely for practical purposes ?

I find it hard to believe that at any point in history the concept of "nothing" or a void was not considered, but there seems to be no record of it. Certainly from a practical perspective the null result was just that, practical, but from the historical record it seems to have also been a reflection of the thinking at the time. It occurs to me that the symbols we use in our writing and mathematics both reflect and influence our thinking, and the number systems necessitating 0 simply had not emerged. If your daily life never includes nothingness, your view of the cosmos is orderly and finite, ruled by gods, then perhaps it really never did emerge.

Then again, we're still learning more about ancient life at that time, and earlier; consider the device I referenced earlier: The Antikythera Mechanism. http://en.wikipedia.org/wiki/Antikythera

Maybe there is more to be discovered, but it seems clear that from an official standpoint the notion of 0 either did not emerge, was considered heretical, or simply preposterous and not worth the consideration. How long ago was Dirac's notion of "the anti-electron" considered to be hogwash, even as cloud chamber photographs were showing evidence of their existence?


Apeiron: Perhaps the best definition of nothingness is the inability to define a context for absence; true nothingness cannot be explored because nothing can exist to explore it without providing context for it. In a physical sense, this may or may not be true, but in every other context it seems inevitable. The dominance of something over nothing makes the imbalance between matter and anti-matter seem trivial by comparison.

 

[apeiron]

Apeiron: Perhaps the best definition of nothingness is the inability to define a context for absence; true nothingness cannot be explored because nothing can exist to explore it without providing context for it. In a physical sense, this may or may not be true, but in every other context it seems inevitable. The dominance of something over nothing makes the imbalance between matter and anti-matter seem trivial by comparison.

The problem with absolute nothing is that it then becomes impossible to explain the existence of something. There is no logical way to say in the beginning was absolutely nothing, then something sprang into being.

But if you instead say in the beginning was a vagueness - a state of infinite potential which is both a nothingness (nothing actually exist locally or globally) and an everythingness (anything could still come into existence because no paths have yet been chosen) - then you have a non-thing that can become a some-thing.

So the argument goes that because there is something (our universe for a start) then the idea of absolute nothingness becomes implausible. Certainly as an initial conditions. Therefore we need to imagine something else that might be as close to a nothing as possible.

Of course, there still remains the question "why did this initial vagueness exist, who caused that?". But then a state of pure potential does not actually "exist", because it just is a formless potential. It is as little like what we mean by existence as it is possible to be.

This is actually the most ancient of ideas. You can see the gist of it in most early creation myths.

In Theogony the initial state of the universe,or the origin (arche) is Chaos, a gaping void (abyss) considered as a divine primordial condition, from which appeared everything that exists. Then came Gaia (Earth) and Eros (Love). Hesiod made an abstraction because his original chaos is something completely indefinite.[6] In the Orphic cosmogony the unageing Chronos produced Aither and Chaos and made a silvery egg in divine Aither. From it appeared the bisexual god Phanes who is the creator of the world.[7]

Some similar ideas appear in the Hindu cosmology which is similar to the Vedic. In the beginning there was nothing in the universe but only darkness and the divine essence who removed the darkness and created the primordial waters. His seed produced the universal germ (Hiranyagarbha), from which everything else appeared.[8]

In the Babylonian creation story Enuma Elish the universe was in a formless state and is described as a watery chaos. From it emerged two primary gods,one male Apsu and one female Tiamat and a third deity who is the maker Mummu and his power is necessary to get the job of birth.[9]. In Genesis the primordial world is described as a watery chaos and the Earth "without form and void". The spirit of the god moved upon the dark face of the waters and created light.[10]

http://en.wikipedia.org/wiki/Theogony

History's first true philosopher, Anaximander of Miletus, was the most systematic developer of the idea (getting away from gods and their spawning progeny - humans only evolving at the end).

And Anaximander called the initial state of infinite, unconstrained, potential, the Apeiron!

The challenge in the modern era is to model this idea of pure potential, of vagueness, with the same mathematical precision we have done for other ontological concepts like nothing and infinity.

Again, various people have worked on this. CS Peirce did the best job IMHO.

Others to dance around the subject have been Max Black (who distinguished vagueness from ambiguity, generality, and indeterminacy), Kortabiński, Adjukiewicz and Fleck (who did not add anything interesting), Karl Menger (who talked about a geometry based on vague objects or “ensembles flous”), Post, Tarski, Knuth and Lukasiewicz (logics of indecision), and most recently, Lotfi Zadeh (fuzzy sets).

My own approach is based on symmetry and symmetry breaking. Vagueness is a state of perfect symmetry, or infinite symmetry. Then it breaks via self-organised criticality. There is a phase transition that develops a nothingness (an unoriented symmetry) to become a something (a realm with scale and direction). So my approach is based on the laws of thermodynamics and the physics of condensed matter.

But anyway, you can see that vagueness, like nothingness and infinity, is a philosophical generalisation about "what exists" that could be, and perhaps should be, framed with a mathematical exactness.

Maybe we need to start by inventing a mathematical symbol for it. Like [?] - the puzzled set :tongue2:.

 

[Pythagorean]

I don't think zero is quite the same as "doesn't exist"

maybe the emtpy set best characterizes existences but

{} != 0

 

[SonyAD]

The concept of zero is a non issue. Claiming irrational numbers are numbers instead of of algorithms is where mathematics lost the plot.

Then it flew over the cuckoo's nest when the concept of complex numbers was concocted.

 

[disregardthat]

Claiming irrational numbers are numbers instead of of algorithms is where mathematics lost the plot.

How is this relevant here though?

Although I would agree in the context of e.g. analysis, I can't say I agree on a general basis. Would you claim that the length of the hypotenuse of a triangle with a right angle between with two sides of length 1 to be an algorithm? Is not this geometric length a length constructed like any rational length?

 

[brainstorm]

"0" is a placeholder in a system of counting that uses place-values to limit the number of symbols for numbers to 9 and 0 (not counting the special numbers like pi, E, i, etc.)

"0" doesn't signify nothingness. It signifies that there are no multiples of 10 in the number 103 except for "100," which of course substitutes a one in the 100s place for something designating "10" in the 10's place, which would be necessary in, say, base 11.

If "0" was a measure of nothingness, it would make sense to measure nothingness in multiples of "0," but that, of course, makes no sense whatsoever. N(0) = 0 always, as does 0^n. It is annoying that ppl obsess over the concept of nothing as if it was something other than a conceptual recognition that a representation/idea of something exists separately from the actual presence of the thing represented.

The primary value of Plato's cave was shelter from the elements.