Is Zero a Real Concept or Just a Metaphysical Idea?

[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 .

 

[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.

 

[BrianConlee]

you have zero apples on the table, yes... but you also have zero cars on the table... zero people... zero unicorns...

You have a "zero" of everything (infinity?) on the table save the air or space if you're in a vacuum.

I have zero unicorns in my house, does that prove they exist?

my reply has zero value btw :( lol

 

[BrianConlee]

oh, and zero times infinity equals ?

that is what you have on the table now...

zero of an infinity of things.

You have nothing of everything all at once.

Zero gives me a headache

 

[GeorgCantor]

Detection events have non-zero, actual values.


Potentiality(zero c60 molecules, just an expectation value) - Actuality(1 real, actual c60 molecule)


Zero is that that which will not take place.

 

[SonyAD]

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.

Why? Isn't accepting irrational numbers as denoting real values like saying you accept infinite complexity in finite space?

I do not even accept the concept of infinity, let alone infinity within the finite.

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?

Yes. sqrt(). With the parameter '2'. Is this not how you define this particular irrational number? Or any other? By the process you 'obtain' it by and the 'seed'?

If it was a number you would define and identify it by its value.

Is not this geometric length a length constructed like any rational length?

No. Imagine a raster image 1024 pixels wide, 768 high. The diagonal is 1024 pixels long. This might be space at the fundamental level. Or it might be a reasonably good analogy.

Of course, this view, that I have, introduces different conundrums. Like how is momentum preserved at the macro scale such that rectilinear uniform motion is possible. Or, how do particles jump from one voxel to another while, on average, preserving their speed and direction? And other stuff I can't remember off the top of my head.

 

[Hurkyl]

If it was a number you would define and identify it by its value.

I wouldn't. I generally have to resort to written strings of symbols, or vocalized strings of phonemes.

There is nothing special about decimal notation that places it above other notations, other than the fact it was drilled into your head as a child.

 

[SonyAD]

I wouldn't. I generally have to resort to written strings of symbols, or vocalized strings of phonemes.

Ok, this is a jab from the sign() thread. I dig it.

There is nothing special about decimal notation that places it above other notations, other than the fact it was drilled into your head as a child.

What other notations? Fractional representation? Mantissa and exponent?

I use fractional representation all the time with symbolic algebra. I've used it for the sign() and H() functions, for instance.

Mantissa and exponent representation is utter fail from every point of view. So sad they explicitly chose the worst possible representation of reals to make standard.

Anyway, except for mantissa & exponent, these conventions are each ways of representing a value down to a fixed granularity (standardised fractional representation as well). The granularity for floating point representation isn't fixed.

 

[alt]

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.



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 Apeiron;

The word though, existed before Anaximander (in order for him to use it - duh, alt) and I believe it also meant then, the in-experienced.

Even today in Greek 'ἄπειρος' (apiros) also means inexperienced, ignorant, as in;

άπειρος υπάλληλος (apiros ipalilos) inexperienced / ignorant employee / servant.

http://www.perseus.tufts.edu/hopper...ic+letter=*a:entry+group=236:entry=a)pei/rwn1

Also, if you;

http://www.thefreedictionary.com/infinite

then go to bottom of page, translations, hit Greek, and you get as the 1st entry ..

1 without end or limits We believe that space is infinite.

But then when you hit the blue link 'άπειρος' on THAT line you get;

άπειρος υπάλληλος - which as I said above, is 'inexperienced employee'.

.. a curious thing at first impressions, I suppose. But it may just be tending to the circular nature of nothingness and infinity - that which is infinite is in(not) experienced.

ά (a); without
πειρά (pira); experience

(I think)

 

[wrongusername]

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 ?

Isn't that more of the English language's fault?

If we take "nothing" to mean a void (as it is used in "A ham sandwich is better than nothing"), then the first statement, meaning the lack of happiness is better than complete happiness in life, does not hold true imo. Thus, the final conclusion does not hold true either, and a conundrum is avoided.

If we also rephrase the first statement to be "Complete happiness in life is better than anything else," then it seems such a conundrum is avoided too because I cannot see how the final conclusion can be reasoned out from the first two statements.

 

[disregardthat]

Sony AD, the length of the hypotenuse occurs as natural as any rational number in euclidean geometry. No one have requested a rational approximation to it, so there is no need to identify it to an algorithm. No one have called the sqrt() algorithm. In analysis, it is different - but in geometry the decimal expansion is irrelevant.

Besides, I imagine a triangle with a irrational side as easy as you imagine a square with rational sides.

 

[Hurkyl]

What other notations?

"sqrt(2)", for example.

Any well-formed symbolic constant expression of type "real number" -- or even a well-formed logical formula with a unique solution -- is a perfectly good way to define and identify a particular number -- and is often times a better way to do so than trying to shoehorn the number into some "standard form" such as its decimal expansion.

 

[apeiron]

(I think)

I think you will find that Greek scholars are happy with the translation of the boundless, the unlimited.

 

[imiyakawa]

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

It's too bad this magical concoction forms the basis of state space in QM and the description of the probabilistic (not in deBB) properties of wavefunctions.

 

[SonyAD]

It's too bad this magical concoction forms the basis of state space in QM and the description of the probabilistic properties of wavefunctions (although not in deBB, of course...)

You have fallen into the trap of believing your representation of reality is reality. The fact that mathematical constructs based on complex numbers can make testable predictions does not make them any more real than infinity or irrational numbers.

"sqrt(2)", for example.

Any well-formed symbolic constant expression of type "real number" -- or even a well-formed logical formula with a unique solution -- is a perfectly good way to define and identify a particular number -- and is often times a better way to do so than trying to shoehorn the number into some "standard form" such as its decimal expansion.

How about you "shoehorn" an irrational "number" into reality. And infinity too, for that matter. See what you end up with.

Not a rational number, by any chance?

Sony AD, the length of the hypotenuse occurs as natural as any rational number in euclidean geometry.

The question has never been whether it occurs in Euclidian geometry. The question has always been whether irrational numbers actually can exist in nature as represented by physical quantities. Which they can't because they don't have values. What is more defining of a number than its value?

This is actually another guise for the question of whether infinity exists. Which it does not.

No one have requested a rational approximation to it, so there is no need to identify it to an algorithm.

The purpose you define them as pairs of seeds(2) and algorithms(sqrt()) for is precisely because you can't assess their value. Because they have none. Because they are not numbers. They are concepts. Symbols of which we can use in calculations as we do symbols in symbolic algebra. When we finish our computations what we end up with is invariably a rational number.

No one have called the sqrt() algorithm. In analysis, it is different - but in geometry the decimal expansion is irrelevant.

Again, this is about whether irrationals are numbers. Which they are not. Because you can obtain numbers through physical measurement. You can not obtain irrationals through physical measurement. Neither can you infinity. They don't exist.

Besides, I imagine a triangle with a irrational side as easy as you imagine a square with rational sides.

Yes. You imagine. Because it doesn't exist.

 

[alt]

Isn't that more of the English language's fault?

If we take "nothing" to mean a void (as it is used in "A ham sandwich is better than nothing"), then the first statement, meaning the lack of happiness is better than complete happiness in life, does not hold true imo. Thus, the final conclusion does not hold true either, and a conundrum is avoided.

If we also rephrase the first statement to be "Complete happiness in life is better than anything else," then it seems such a conundrum is avoided too because I cannot see how the final conclusion can be reasoned out from the first two statements.

I cannot see how the final conclusion can be reasoned out from the first two statements.

Yes, absurd, isn't it ? Of course it can't. As you said, an English language fault, or perhaps, more the point, the circular, maybe dual nature of the word / concept 'nothing'

Which is why I'm making (or trying to make) a little foray into another, older language, to see if anything fruitful can come from that. But I'm having similar problems it seems.

 

[imiyakawa]

You have fallen into the trap of believing your representation of reality is reality. The fact that mathematical constructs based on complex numbers can make testable predictions does not make them any more real than infinity or irrational numbers

I'm not proposing that they're real. I'm going to stop posting in this thread because my QM and mathematics is quite shocking.

I was under the impression that the ratios of complex numbers are required to formalize what is occurring during state to state transitions of superposed wavefunctions. What does this tell us?

|a> -> |a'>, and |b> -> |b'>
But |a> + |b> does not -> |a'> + |b'>

You need a complex number z
z1|a> + z2|b> -> z1|a'> + z2|b'>

Does this tell us anything significant about complex numbers?

 

[disregardthat]

Sony AD, this has nothing to do with physical quantities, but has everything to do with euclidean geometry. I brought it up as an example of a piece of mathematics in which irrational numbers are used as naturally as rational numbers. What can occur and what cannot occur in nature is completely irrelevant to mathematics. All triangles are imagined, both those with irrational sides and those with rational sides. It has nothing to do with physical measurement, that is a blind road. Mathematics is a field concerning abstractions, not physical measurement. And it is certainly not bounded by whatever physical perspective one might have.

You say we can't assess their value, but this is a play with words. sqrt(2) is fine as it is, and I can compare it to any given rational or irrational number if I want. A value is not necessarily bound to a representation through rationals which you seem to imply. Treating irrationals symbolically is completely justified if it can be done so consistently, and it can.

Numbers can be abstracted in various ways.

 

[alt]

I think you will find that Greek scholars are happy with the translation of the boundless, the unlimited.

I'll take your word on that, as I'm not familiar with recent Greek scholars views on it.

From my memory of Aristotle and others who had considered 'apeiron' though, I recall there was never a precise, or even a commonly accepted definition - I recall something about 'primordial chaos' ? And other divergant views ? I can reseach it later if neccessary.

In any case, my enquiry here (such as it is), is not so much about translation. I merely used translation to find 'apeiron' in the Greek, then seek it's meaning.

Anaximander used the Greek available to him, to describe what he thought was an infinite, unlimited, ageless mass. And he used;

a (not)
peira (experienced)

NOT EXPERIENCED

Maybe this is a trivial point to you. I was hoping you might have a view as to how one gets from 'not experienced' to 'infinite, unlimited, ageless mass'.

And don't forget, we were discussing 'zero' (miden) on this thread, which you developed to 'apeiron'. I'm glad you did, but I'm as confused as ever.

 

[nismaratwork]

I'll take your word on that, as I'm not familiar with recent Greek scholars views on it.

From my memory of Aristotle and others who had considered 'apeiron' though, I recall there was never a precise, or even a commonly accepted definition - I recall something about 'primordial chaos' ? And other divergant views ? I can reseach it later if neccessary.

In any case, my enquiry here (such as it is), is not so much about translation. I merely used translation to find 'apeiron' in the Greek, then seek it's meaning.

Anaximander used the Greek available to him, to describe what he thought was an infinite, unlimited, ageless mass. And he used;

a (not)
peira (experienced)

NOT EXPERIENCED

Maybe this is a trivial point to you. I was hoping you might have a view as to how one gets from 'not experienced' to 'infinite, unlimited, ageless mass'.

And don't forget, we were discussing 'zero' (miden) on this thread, which you developed to 'apeiron'. I'm glad you did, but I'm as confused as ever.

That which is not experienced, is not the same as "inexperienced" in the sense of naive. You could read that as "that which cannot be experienced", or a number of other ways. I would also like the namesake of this sidetrack to weigh in. It may be that the usage in this case was meant to imply everything outside of the human experience, which is "boundless". I don't know, I'm not a scholar of ancient Greek either, but I wouldn't mind delving into it. Apeiron, I for one am not hostile to your interpretation, and I don't think that alt is either. I really would like to hear your take on this; I'm sure you understand the difference between a direct translation and the meaning at the time and later.

 

[Hurkyl]

How about you "shoehorn" an irrational "number" into reality.

What does this have to do with what I was saying.

I'm not sure how to make sense of your comment, though, since irrational numbers and infinitary methods are used of all of the best physical theories we have of reality.

Or rather, I have a pretty good idea what argument you're implying, but have never understood why people think "la la la, I can pretend everything is a natural number and still function in society" is a convincing argument of, well, whatever point they are trying to convince people of with it.

 

[Jimmy Snyder]

I haven't read but a few of the posts here so I'm sorry if this has already been said. It has always interested me that there is a difference when comparing one apple to one orange that doesn't have a counterpart when comparing zero apples to zero oranges.

 

[Pythagorean]

I haven't read but a few of the posts here so I'm sorry if this has already been said. It has always interested me that there is a difference when comparing one apple to one orange that doesn't have a counterpart when comparing zero apples to zero oranges.

I think the difference is still there. If you have 1 apple, but you're counting oranges, then you have zero oranges, but if you were to count apples you would have 1 apple.

You're looking at the limited case where there is zero fruits to make your claim from, but that doesn't encompass the case where there is only one kind of fruit.

 

[Pythagorean]

What does this have to do with what I was saying.

I'm not sure how to make sense of your comment, though, since irrational numbers and infinitary methods are used of all of the best physical theories we have of reality.

Or rather, I have a pretty good idea what argument you're implying, but have never understood why people think "la la la, I can pretend everything is a natural number and still function in society" is a convincing argument of, well, whatever point they are trying to convince people of with it.

You can blame my church for that.


(hint: see name)

 

[apeiron]

Maybe this is a trivial point to you. I was hoping you might have a view as to how one gets from 'not experienced' to 'infinite, unlimited, ageless mass'.

The usual take on the entymology is that the root term is peras - limit. Hence the unlimited.

But regardless of entymology, the meaning of words lies in their use, their semiosis. So it was how the ancients used the term that really gives apeiron its meaning.

 

[my_wan]

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

maybe the emtpy set best characterizes existences but

{} != 0

At a fundamental level it seems to me that this debate is an existential infinitesimal or an element lacking existence. If we take the above academic definition, then 0 is empty while {} is an existential infinitesimal with an empty interval in space. Recursivity actually allows us to avoid any direct conflict between 0 and {}. Simply refer to 0 as lacking a thing, whereas {} is an empty thing. Yet 0 can become an {} by defining a set of all things lacking a thing. Thus the other threads asking if nothing actually exist. The academic resolution merely avoids the issue by recursivity.

My personal take on it is that 0 and {} effectively equivalent in many ways, except that {} is existential, whereas 0 is merely a potential, a degree of freedom of an {}, depending on relative cardinalities. Though even that is not absolute from a fundamental coordinate independent perspective. This identifies an {} as an infinitesimal with an unknown transfinite cardinality, a limit. Thus an empty set is only empty in the sense that its interval in space is empty, whereas 0 is empty at that point, or at least, more generally, empty of a member of the set in question.

It's weird, but as soon as you accept transfinite, things get weird. I conceptually prefer non-standard calculus, but that appears to have required the limit approach to derive any useful meaning.

 

[disregardthat]

Is it just me, or is this discussion getting very ambiguous? How can you even talk about a 'direct conflict' between 0 and {}? How could such a conflict ever arise, and in what terms would it manifest itself? I can honestly not understand how formal symbols like {} and 0 subject to well-defined formal operations can cause any confusion at all!

A sentence like

"This identifies an {} as an infinitesimal with an unknown transfinite cardinality, a limit."

is completely nonsensical to me. You appear to have arbitrarily thrown three different mathematical concepts (infinitesimal, transfinite cardinality, limit) into a grammatically correct sentence, but nothing more!

{} (empty-set) is neither an infinitesimal, nor a limit. And its cardinality is 0; not unknown and not transfinite.

The previous post strikes me (and this might be caused by my lack of understanding of your terms and reasoning) as completely incoherent.