Poll: Does zero exist? - intellectual debate
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The base 10 system exists as well as the base 60 system, all of these are within the field of mathematics. The system is abstract but it is not arbitrary. It is a complete system that is true to itself.AC10 said:
Just because something is not real does not mean that it doesn't exist. Consider measurement, if no-one is around to measure a distance does that mean that there is no such thing as distance? If you die, does mathematics cease to exist? If everyone just disappeared all of a sudden would mathematics suddenly be false?
So mathematics exists and is tangible but has no physical presence. Yet physical presence is irrelevant, what matters is that it isn't arbitrary, that it is a true and self-fulfilling system.
:O i think ppl have got a bit off track and in some ways I agree - especially with the fact that 0 exists as a representation of nothing - however 1/3 the arguments here say you can't divide by zero = and if that is true which everyone says - give me the reason why! - which one of you bothered to do!
However the crux of this whole thread was meant to be - although 0 is used in mathematical formula does that mean it exists in the real sense - if you look at our universe as an example or at least as how modern physicists see this many believe in constant expansion theory i.e. big bang flung matter in all directions from one initial point and they are accelerating away from each other (see dopler effect). Now If you assume we are the only think in existence (lets call that our universe - our galaxies all in cluster accelerating away from each other) and the 'outside' of our galaxy is nothing but also expanding infinitely (i.e. we do not live inside a giant box) - see string theory for explanation about there being 'no outside' - thou I am a bit of a new comer to string theory so my example may be null. Any way if we exist something made us happen - God or something which means we were made from something lets call this something value N. Now if there is nothing so 0 lets call it then 0 + something = us or creation, or we could call it, existence of something + cause => effect + time => now, this implies something existed with 0 for us to be made or we are still unable to understand 0 and its full properties or why it has to follow a set of rules. This question was more aimed at asking does 0 exist in the real sense of the world rather than in purely mathematics and looking for an answer to why you ant divide by 0
However the crux of this whole thread was meant to be - although 0 is used in mathematical formula does that mean it exists in the real sense - if you look at our universe as an example or at least as how modern physicists see this many believe in constant expansion theory i.e. big bang flung matter in all directions from one initial point and they are accelerating away from each other (see dopler effect). Now If you assume we are the only think in existence (lets call that our universe - our galaxies all in cluster accelerating away from each other) and the 'outside' of our galaxy is nothing but also expanding infinitely (i.e. we do not live inside a giant box) - see string theory for explanation about there being 'no outside' - thou I am a bit of a new comer to string theory so my example may be null. Any way if we exist something made us happen - God or something which means we were made from something lets call this something value N. Now if there is nothing so 0 lets call it then 0 + something = us or creation, or we could call it, existence of something + cause => effect + time => now, this implies something existed with 0 for us to be made or we are still unable to understand 0 and its full properties or why it has to follow a set of rules. This question was more aimed at asking does 0 exist in the real sense of the world rather than in purely mathematics and looking for an answer to why you ant divide by 0
Something you all have failed to realize is that none of the numbers exist.
Yes. That's right. Not 0, nor 1, nor -2, nor 893, 2.445 or 3.14159265349798...
You see, all mathematics is simply a model. That's why we have things like an endless decimal in pi and e, as well as other irrationals. In fact, 0 is somewhat of an idealistic number. It's a member of the set we call Rationals, defined as all numbers that can be expressed as a fraction of 2 integers (including 0 and negatives, with the exception of not 0 in the denominator, of course). The fact that we can express these numbers so succinctly is evidence to the fact that it's inherently inaccurate. Maybe just a smidgen, but it is. In fact, you will never encounter a single rational number in the real world in your entire existence. It can be proven mathematically. Irrational numbers are infinitely more dense on the number line than Rationals. Therefore, there is probability 1 that a number picked at random will be Irrational. The universal constants that we have to work around, such as pi and Euler's number (both of which are Irrational), have no great expressions in our number representation system, and this is due to that simple fact: all numbers are intellectual concepts created by man to help him understand what's going on around him. No, you can't divide by 0, but that's not 0's fault. It's man's fault for using such a problematic system.
I'll illustrate this using fruit. Let's say I have 4 apples. If I know I have 4 apples, that means I am using the concept of the number 4 to help me understand how many apples there are. If your definition of existence is that it lives in the minds of man (see the Platonic Forms, and the third tier of his pyramid--first the Good, then the Forms, then mathematical and grammatical concepts), then yes, Four exists. And so does Zero, and so does Grandelflfljjjssshargensmarg, because I just made it up (it means a very big sneeze). If your definition of existence is that its existence (or nonexistence) does not derive from its perception (or lack of perception) in the minds of those perceiving it, then no, none of the numbers exist. And neither does the concept of school, the internet, or government.
Let's think of it this way: If I have 4 apples, and I can look at them and see that there are 4. But let's say we don't exist. There is no one there to perceive there being 4 apples. How could the concept of 4 still exist, without someone to understand it? Each apple exists, yes, but it takes a point of view to see there being 4.
Still don't believe me? Well even if you don't agree with the rest of my statements, you have to agree that numbers are inherently inaccurate. Saying "I have 4 apples" is fine for conversation, but that leaves out important bits of information:
Are they all Granny Smith Apples?
Yes. I have 4 Granny Smith Apples.
OK, well are they all ripe?
Yes. I have 4 ripe Granny Smith Apples.
OK, well did they all come from the same tree?
Yes. I have 4 ripe Granny Smith Apples from the second tree on the left as you come to the Johnsons' orchard.
...
And the conversation could continue, to the point that they obviously cannot occupy the same space, and be made up of the same atoms. This proves that unless you describe each apple individually (without a concept of numbers), you cannot include all the information, even using the concept of the whole number 4 applied to a fairly simple-looking real-world problem.
My point: numbers do not exist.
Yes. That's right. Not 0, nor 1, nor -2, nor 893, 2.445 or 3.14159265349798...
You see, all mathematics is simply a model. That's why we have things like an endless decimal in pi and e, as well as other irrationals. In fact, 0 is somewhat of an idealistic number. It's a member of the set we call Rationals, defined as all numbers that can be expressed as a fraction of 2 integers (including 0 and negatives, with the exception of not 0 in the denominator, of course). The fact that we can express these numbers so succinctly is evidence to the fact that it's inherently inaccurate. Maybe just a smidgen, but it is. In fact, you will never encounter a single rational number in the real world in your entire existence. It can be proven mathematically. Irrational numbers are infinitely more dense on the number line than Rationals. Therefore, there is probability 1 that a number picked at random will be Irrational. The universal constants that we have to work around, such as pi and Euler's number (both of which are Irrational), have no great expressions in our number representation system, and this is due to that simple fact: all numbers are intellectual concepts created by man to help him understand what's going on around him. No, you can't divide by 0, but that's not 0's fault. It's man's fault for using such a problematic system.
I'll illustrate this using fruit. Let's say I have 4 apples. If I know I have 4 apples, that means I am using the concept of the number 4 to help me understand how many apples there are. If your definition of existence is that it lives in the minds of man (see the Platonic Forms, and the third tier of his pyramid--first the Good, then the Forms, then mathematical and grammatical concepts), then yes, Four exists. And so does Zero, and so does Grandelflfljjjssshargensmarg, because I just made it up (it means a very big sneeze). If your definition of existence is that its existence (or nonexistence) does not derive from its perception (or lack of perception) in the minds of those perceiving it, then no, none of the numbers exist. And neither does the concept of school, the internet, or government.
Let's think of it this way: If I have 4 apples, and I can look at them and see that there are 4. But let's say we don't exist. There is no one there to perceive there being 4 apples. How could the concept of 4 still exist, without someone to understand it? Each apple exists, yes, but it takes a point of view to see there being 4.
Still don't believe me? Well even if you don't agree with the rest of my statements, you have to agree that numbers are inherently inaccurate. Saying "I have 4 apples" is fine for conversation, but that leaves out important bits of information:
Are they all Granny Smith Apples?
Yes. I have 4 Granny Smith Apples.
OK, well are they all ripe?
Yes. I have 4 ripe Granny Smith Apples.
OK, well did they all come from the same tree?
Yes. I have 4 ripe Granny Smith Apples from the second tree on the left as you come to the Johnsons' orchard.
...
And the conversation could continue, to the point that they obviously cannot occupy the same space, and be made up of the same atoms. This proves that unless you describe each apple individually (without a concept of numbers), you cannot include all the information, even using the concept of the whole number 4 applied to a fairly simple-looking real-world problem.
My point: numbers do not exist.
claymorez, you ask why no one can divide by 0. You yourself proved this in your initial post. There are rules in math, and they must always hold true. Sometimes this necessitates exceptions.
For any integer n, it is true that
n*0 = 0
It is also true that
n/n = 1
It is also true that one can perform equal operations on both sides of an equation without affecting its validity.
If we start with
n*0 = 0
and divide both sides by 0, we end up with
n = 0/0
If we say that
0/0 = 1
then
n = 1
which is a problem, because we have given in the beginning that n may be any integer.
There are 2 options to alleviate this difficulty:
1. Make an exception for these operations on 0.
2. Make the domain of n limited to 1.
Obviously, the more robust and useful approach is the former one. As it turns out, there are many exceptions for the domains of functions, such as logarithms (positive numbers only), inverse trig functions (-1,1), and others. Again, this goes back to my point that numbers operate at the beck and call of the physical world, not the other way around, no matter how much we may think so. For example: you may find it really cool to learn that
e^pi*i = -1
where e is euler's number, pi is the ratio of a circle's circumference to its diameter, and i is the square root of -1. But this is not because math is so awesome that it makes the very natural world bend to its might. It's because the natural world is real and immutable, while mathematics is a derived concept, subject to exceptions and holes in logic or continuity. I think you are overthinking this question. My question to you, if you still have problems with my answer, is this:
Do Good, Love, Wisdom, Sin, Power and Thought exist?
I know this is sort of off topic, but I think it will help us answer your question. Tell me your explicit definition of "existence," then think of each one individually, and answer for them individually or in groups, however you think they exist or don't exist.
For any integer n, it is true that
n*0 = 0
It is also true that
n/n = 1
It is also true that one can perform equal operations on both sides of an equation without affecting its validity.
If we start with
n*0 = 0
and divide both sides by 0, we end up with
n = 0/0
If we say that
0/0 = 1
then
n = 1
which is a problem, because we have given in the beginning that n may be any integer.
There are 2 options to alleviate this difficulty:
1. Make an exception for these operations on 0.
2. Make the domain of n limited to 1.
Obviously, the more robust and useful approach is the former one. As it turns out, there are many exceptions for the domains of functions, such as logarithms (positive numbers only), inverse trig functions (-1,1), and others. Again, this goes back to my point that numbers operate at the beck and call of the physical world, not the other way around, no matter how much we may think so. For example: you may find it really cool to learn that
e^pi*i = -1
where e is euler's number, pi is the ratio of a circle's circumference to its diameter, and i is the square root of -1. But this is not because math is so awesome that it makes the very natural world bend to its might. It's because the natural world is real and immutable, while mathematics is a derived concept, subject to exceptions and holes in logic or continuity. I think you are overthinking this question. My question to you, if you still have problems with my answer, is this:
Do Good, Love, Wisdom, Sin, Power and Thought exist?
I know this is sort of off topic, but I think it will help us answer your question. Tell me your explicit definition of "existence," then think of each one individually, and answer for them individually or in groups, however you think they exist or don't exist.