Do numbers exist?

[bunny]

I mean in the platonic sense - separate from the physical universe, eternal, timeless, unchanging, etc etc. Or are they purely constructs of language, relationships between physical things, something else?

I'd be interested in people's opinions - it seems the only thing I still find mysterious.

[hmkpoker]

Numbers (1,2,3,4) are just constructs of language. That which they represent, that is, logical operators (yes/no) are the cornerstone of thinking.

[bunny]

So why are they so "unreasonably effective" at describing the physical world? If they are the product of intelligent beings, why should they work so well at modelling reality? (Not in the simple 1 apple plus 1 apple = 2 apples kind of way, but in the more hardcore quantum physics kind of way).

[FortunaMaximus]

They're just a way to track and analyze rational flux.

1 is to 1 as 1 is to 1

2 is to 2 as 1 + 1 is to 2.

Of course, it gets a lot more complicated than that. But that's a way to describe rational growth.

0 is not nothing. It's the ultimate placeholder.

<shrugs> You know, I'm not sure either. 2 + 2 = 5 if 2 is 2.5 in an algebraic format.

Why do you ask?

Edit: Pretty sure it occurred '98ish, I can detect it, administer it, tell the difference, but I think it's a multiple-emergent.

[livin_a_lie]

[ QUOTE ]
Why do you ask?

[/ QUOTE ]

[ QUOTE ]
it seems the only thing I still find mysterious.

[/ QUOTE ]

He's almost done!

[Mickey Brausch]

[ QUOTE ]
Do numbers exist? I mean in the platonic sense - separate from the physical universe, eternal, timeless, unchanging, etc etc. Or are they purely constructs of language, relationships between physical things?

[/ QUOTE ]You can't see, hear, or smell numbers in the physical world*. It is unlikely that you might physically bump into 8501 or "square root of -1" any time soon.

But numbers do exist. I mean in the platonic sense - separate from the physical universe; eternal, timeless, unchanging, etc. They are purely constructs of language, denoting relationships between physical things.

They will go away when we'll go away.

Mickey Brausch

---

* The "visualization", and other physical manifestations, of numbers as experienced by mathematical geniuses --and idiots savant-- notwithstanding.

[siegfriedandroy]

[ QUOTE ]
it seems the only thing I still find mysterious.

[/ QUOTE ]

amazing

[Magic_Man]

There will always be countable amounts of things in any universe. 2 of something is always twice as many as 1 of the same thing. In that sense, numbers exist independent of the physical univers.

~MagicMan

[Lestat]

Physical (and non-physical) objects exist. The math used to count them is entirely a man-made concept. Kind of like asking if letters, words, or language really exist in a different context than we understand them. (I think).

[FortunaMaximus]

I feel 'em, see them, walk on them.

A tree's just a structure of numbers. A single tree can be older than the Universe. Blueshift it if only in your imagination, and look up in the sky.

Goddamn.

[Magic_Man]

The words we use to describe them are certainly a man-made concept, but are the numbers themselves? I've been thinking a lot about this lately, coincidentally. I was wondering what intelligent alien civilizations' math would look like. I have to assume that every intelligent creature would "discover" numbers eventually, and that they'd know about pi, e, the pythagorean theorem, and lots of other things familiar to us. The concept of there being 1 or something, or 2 of something, or 5298 or something, is independent of who/what is thinking about it.

[FortunaMaximus]

Don't be so sure.

They'd discover the 0, 1, few, many, but the interpretations can and will be drastically variant.

Silicates might perceive a count along the color spectrum, for instance. Or taste by atomic masses.

Yeah.

[Lestat]

[ QUOTE ]
The words we use to describe them are certainly a man-made concept, but are the numbers themselves? I've been thinking a lot about this lately, coincidentally. I was wondering what intelligent alien civilizations' math would look like. I have to assume that every intelligent creature would "discover" numbers eventually, and that they'd know about pi, e, the pythagorean theorem, and lots of other things familiar to us. The concept of there being 1 or something, or 2 of something, or 5298 or something, is independent of who/what is thinking about it.

[/ QUOTE ]

But why can't the same be said for language? Just as every experience and thing can be broken down into words, so too can everything be broken down into math. If an intelligent alien saw a quasar, he would somehow be able to describe what he saw, which may be the exact same thing as you saw and describe to me. In other words, what happened - happened. Same for the alien as you. So if he saw 3 stars, why wouldn't "3" exist for him (whatever he calls it), the same as for you? In other words, the number "3" must exist, same as the 3 stars exist.

btw- this is way out of my realm. I'm terrible with numbers and just spewing thoughts. Don't anyone take me seriously when it comes to math.

[aeest400]

[ QUOTE ]
Numbers (1,2,3,4) are just constructs of language. That which they represent, that is, logical operators (yes/no) are the cornerstone of thinking.

[/ QUOTE ]

Not even in the ballpark.

[David Sklansky]

Are you aware that you have asked the only question that mathmeticians all debate?

[bunny]

[ QUOTE ]
Are you aware that you have asked the only question that mathmeticians all debate?

[/ QUOTE ]
Well I remember debating it in the tea-room at lunch. I knew it was widespread, didnt know it was universal. What's your opinion?

[bunny]

[ QUOTE ]
They are purely constructs of language, denoting relationships between physical things.

They will go away when we'll go away.

Mickey Brausch


[/ QUOTE ]
If we look back in time to before intelligence existed (ie rotations of proto-galaxies or somesuch) they followed things like the inverse square law. Doesnt it seem like numbers existed then, even though nobody was around to think of them? Why should the numbers go away when we do (which also seems to contradict the timeless, eternal comment you made?) I cant see why they are dependant on us - Pythagoras' theorem was true before anyone thought of it.

[bunny]

[ QUOTE ]
The words we use to describe them are certainly a man-made concept, but are the numbers themselves? I've been thinking a lot about this lately, coincidentally. I was wondering what intelligent alien civilizations' math would look like. I have to assume that every intelligent creature would "discover" numbers eventually, and that they'd know about pi, e, the pythagorean theorem, and lots of other things familiar to us. The concept of there being 1 or something, or 2 of something, or 5298 or something, is independent of who/what is thinking about it.

[/ QUOTE ]
This is the heart of my pondering too. The maths that different cultures came up with had different standards of proof, different focuses and axioms/areas of investigation. But they were all consistent.

When you actually do mathematics it doesnt seem like you have any choice. You choose your axioms and definitions, obviously, but the consequences of that are not up to you - after that it feels like you are discovering properties about real objects.

The various materialist philosophies of mathematics struggle with this, it seems to me.

[bunny]

[ QUOTE ]
[ QUOTE ]
it seems the only thing I still find mysterious.

[/ QUOTE ]

amazing

[/ QUOTE ]
*shrug* Perhaps I should be clear that there's a lot of stuff that I realise I dont know how it works - but not much that actually baffles me as to how it could possibly be. Maths and knowledge of maths by physical beings is a mystery I have no handle on at all.

[bunny]

[ QUOTE ]
But why can't the same be said for language? Just as every experience and thing can be broken down into words, so too can everything be broken down into math. If an intelligent alien saw a quasar, he would somehow be able to describe what he saw, which may be the exact same thing as you saw and describe to me. In other words, what happened - happened. Same for the alien as you. So if he saw 3 stars, why wouldn't "3" exist for him (whatever he calls it), the same as for you? In other words, the number "3" must exist, same as the 3 stars exist.

btw- this is way out of my realm. I'm terrible with numbers and just spewing thoughts. Don't anyone take me seriously when it comes to math.

[/ QUOTE ]
There seems a difference between language and maths to me. Language seems arbitrary and not just in the obvious way of which sounds are connected to which concepts, etc. I mean more fundamentally in that we didnt have to use nouns, verbs, etc - it's just how we chose to do it. We had no such choice with whether 28 is a perfect number though - once we choose to look at perfect numbers (ie define the term) we see that 28 is perfect and 30 isnt.

PS - I am aware, perhaps even hoping, that the deficiency may well be in my powers of conception - perhaps they are fundamentally the same thing. All I can say is it doesnt seem like it if you do any real maths.

[Turn Prophet]

Read some Frege. Then Russell. Then Hilbert. Then Godel.

Regrettably, none of them could come up with convincing answers to this problem (worse yet, Frege backed himself into a corner). Nor have philosophers since Plato come up with a convincing solution to the problem of universals/ideas/concepts.

As it turns out, idealism (the notion that concepts are entirely mental constructs) seems like the most convincing scheme in our current scientific (ie materialist) paradigm, but it unfortunately runs into a few problems. For instance, are relations real (ie physical) things? Probably not. But would Mt. Everest cease to have the property "being taller than" a one foot anthill if there were no cognitive beings? Uh oh...

Regardless of its problems, I'd say idealism is still the best we can do with numbers and other conceptual constructs. It still remains a major problem in Metaphysics, the Philosophy of Language, and Logic and Mathematics.

[chezlaw]

[ QUOTE ]
Are you aware that you have asked the only question that mathmeticians all debate?

[/ QUOTE ]
I know a lot of professional mathematicians and there's nothing they are less interested in debating though much they are equally uninterested in debating.

Whether the next conference should be in Brazil or Nice seems to fascinate them.

chez

[David Sklansky]

Amazing. This is the second time someone decided to nitpick my use of the word "debate" instead of "disagreement".

[chezlaw]

[ QUOTE ]
Amazing. This is the second time someone decided to nitpick my use of the word "debate" instead of "disagreement".

[/ QUOTE ]
they don't disagree either, they never even think about it and hold no views to disagree about.

You mean if they did hold views they would disagree?

chez

[David Sklansky]

Yes they do. Ask them. Almost all mathmeticians count theselves as Platonists or Formalists or occasionaly Constuctionists. But they don't talk about it much.

[yukoncpa]

[ QUOTE ]
They are purely constructs of language, denoting relationships between physical things.

They will go away when we'll go away.

Mickey Brausch


[/ QUOTE ] Mickey, are you saying that relationships between physical things will go away when we go away or that constructs of language will go away or both? Just confused. Are numbers and relationships between physical things the same thing?

[chezlaw]

[ QUOTE ]
Yes they do. Ask them. Almost all mathmeticians count theselves as Platonists or Formalists or occasionaly Constuctionists. But they don't talk about it much.

[/ QUOTE ]
They really don't. The question is not a mathematical one and most have no interested in hi falootin stuff.

More importantly, the answer is Brazil - Brazil.

chez

[David Sklansky]

"They really don't. The question is not a mathematical one and most have no interested in hi falootin stuff."

They used to. But I hope you are right.

[chezlaw]

[ QUOTE ]
"They really don't. The question is not a mathematical one and most have no interested in hi falootin stuff."

They used to. But I hope you are right.

[/ QUOTE ]
/images/graemlins/grin.gif

[siegfriedandroy]

YEAH BABAAAAAHHHH

[siegfriedandroy]

I REMember the first time i got kicked out of horseshoe when i was 21

[siegfriedandroy]

i also remember the snd time, when i was 21

[Lestat]

<font color="blue"> All I can say is it doesnt seem like it if you do any real maths. </font>

Alas, I am very inept at math. I do very much appreciate its beauty though. I can imagine how a mathemetician finds joy and glorious satisfaction in solving a complex math problem almost like a musician might feel about creating a particular piece of music. Perhaps more so as he has discovered a new sense or meaning.

But I still say things like numbers, music, art, language are all there for the taking. They are physically present in the universe and it's how man/alien chooses to use and interpret them. But like stated, mine is a very sophomoric opinion when it comes to math.

[CityFan]

[ QUOTE ]
I mean in the platonic sense - separate from the physical universe, eternal, timeless, unchanging, etc etc. Or are they purely constructs of language, relationships between physical things, something else?

I'd be interested in people's opinions - it seems the only thing I still find mysterious.

[/ QUOTE ]

For me, yes, although I really think it's a question of definition rather than deep philosophy.

Under my working definition of the verb "to exist", numbers exist. That defintion of "to exist" is quite broad though - if all mathematical constructs exist then:

a) The "existence" of the universe is inevitable, because it exists as a mathematical possibility.
b) All other possible universes also exist.
c) Universes with and without a "God" exist...
d) Etc. etc., you can see where this is leading.
e) Quantum theory is great.

[CityFan]

[ QUOTE ]
Read some Frege. Then Russell. Then Hilbert. Then Godel.

Regrettably, none of them could come up with convincing answers to this problem (worse yet, Frege backed himself into a corner). Nor have philosophers since Plato come up with a convincing solution to the problem of universals/ideas/concepts.

As it turns out, idealism (the notion that concepts are entirely mental constructs) seems like the most convincing scheme in our current scientific (ie materialist) paradigm, but it unfortunately runs into a few problems. For instance, are relations real (ie physical) things? Probably not. But would Mt. Everest cease to have the property "being taller than" a one foot anthill if there were no cognitive beings? Uh oh...

Regardless of its problems, I'd say idealism is still the best we can do with numbers and other conceptual constructs. It still remains a major problem in Metaphysics, the Philosophy of Language, and Logic and Mathematics.

[/ QUOTE ]

My advice would be to read some elementary set theory, at least up to the construction of the natural numbers from the empty set.

Then make your own mind up whether these things "exist".

[lastcardcharlie]

Four points:

(1) Calculus demonstrates a profound connection between number and space, or between algebra and geometry. Space is composed of numbers and numbers are to be considered as elements of an ambient space. The ontology of numbers is tied to the ontology of space.

(2) Numbers are not to be thought of in isolation but as elements of a number system, namely a collection of elements on which operations such as addition, subtraction, multipication and division are defined. It is these operations that are the basic subject matter of algebra and to ask whether numbers exist is to this extent to ask whether addition, etc. exists.

(3) It is sometimes overlooked that there are very different number systems commonly in use. We use a completely different number system (modular arithmetic) to tell the time, for example, than the one we learn in school arithmetic. Again, numbers do not exist independently of an ambient number system, and there is certainly more than one of these.

(4) There is a profound connection between number and logic. The only way I can understand logic is to see it as something going back to the mists of time, like music. Asking whether logic is independent is a little bit like asking whether music is independent, in my opinion.

[madnak]

I believe they're real, but not in the Platonic sense. I do think certain ratios are "pinned" to the fabric of reality. Perhaps they're properties of basic physical units, or some such thing.

This is very much a question of pure philosophy.

[bunny]

[ QUOTE ]
[ QUOTE ]
Read some Frege. Then Russell. Then Hilbert. Then Godel.

Regrettably, none of them could come up with convincing answers to this problem (worse yet, Frege backed himself into a corner). Nor have philosophers since Plato come up with a convincing solution to the problem of universals/ideas/concepts.

As it turns out, idealism (the notion that concepts are entirely mental constructs) seems like the most convincing scheme in our current scientific (ie materialist) paradigm, but it unfortunately runs into a few problems. For instance, are relations real (ie physical) things? Probably not. But would Mt. Everest cease to have the property "being taller than" a one foot anthill if there were no cognitive beings? Uh oh...

Regardless of its problems, I'd say idealism is still the best we can do with numbers and other conceptual constructs. It still remains a major problem in Metaphysics, the Philosophy of Language, and Logic and Mathematics.

[/ QUOTE ]

My advice would be to read some elementary set theory, at least up to the construction of the natural numbers from the empty set.

Then make your own mind up whether these things "exist".

[/ QUOTE ]
This is exactly where I am coming from (although my field was finite projective geometry - the same feeling though). I would defy anyone doing pure maths to feel they were inventing or exploring linguistic conventions of some sort rather than exploring and discovering facts about real "objects".

Thanks for the opinions to all (although chezlaw and DS still havent answered the actual - I'd be curious what the two of you thought as well).

[bunny]

[ QUOTE ]
[ QUOTE ]
Yes they do. Ask them. Almost all mathmeticians count theselves as Platonists or Formalists or occasionaly Constuctionists. But they don't talk about it much.

[/ QUOTE ]
They really don't. The question is not a mathematical one and most have no interested in hi falootin stuff.


[/ QUOTE ]
Although it may be true that they dont argue about it actively much, I expect this is because they consider it unresolvable. Every mathematician in my department was at least familiar with the 3 different positions DS refers to and all had some opinion, no matter how half-hearted.

[CityFan]

The reason I said set theory is because that will show directly how "numbers" can be constructed from the most basic logical concepts.

Unfortunately I don't have a "field" as I only ever did a bachleor's degree, but am considering strongly going back to do a PhD.

Where did you study? I'm looking to find out a bit more about the best US universities for maths. I'm UK based, but I like the US.

[Matt R.]

In my opinion, the moment you create a countable system, numbers absolutely exist. Take away that system, and numbers lose the basis for their definition. They lose all meaning, and thus they cease to exist.

Therefore, in the context of our physical reality, numbers must exist in a real sense (the same way any idea exists). I guess my question would be -- is it possible to create a non-countable system? I can't imagine one -- but that may be because I lack the imagination.

I think this question ultimately boils down to what "reality" (for lack of a better word), is like outside of the physical. If this even exists, of course. Because we cannot know the nature of something without the groundings of our known reality, I think that the question itself is unanswerable.

[FortunaMaximus]

MIT, Berkeley, USC are the obvious choices.

If you're looking to get over the pond, but not necessarily the States, the University of Toronto trumps 'em all. And I ain't sayin' that 'cause I'm a Toranna boy.

[luckyme]

[ QUOTE ]
But would Mt. Everest cease to have the property "being taller than" a one foot anthill if there were no cognitive beings? Uh oh...


[/ QUOTE ]

Instant cessation. "taller than" is only derived from a measurement from one arbitray point to another. Measured from earth center it is ( also, earth center is only a special point in the universe because I decide to define it), measured from the moons orbit it isn't.

How much of that applies to 'real math'likely just depends on what we decide 'exist' means today, but I'd be very suspicious.

luckyme

[FortunaMaximus]

Quantum principle:

Observer effect, right. If you can think of it and create it, it exists and always will.

[samsonite2100]

Would a super-intelligent race of insectoid aliens have the same conception of numbers as humans? Maybe, maybe not. If they were highly predisposed to hive-mind thinking, they might be unable to distinguish single entities and instead see things as part of larger groups.

I know that example is kind of weak, but I do think there's an inevitable prejudice towards thinking that our modern human conception of numbers is the only possible conception.

[FortunaMaximus]

Yeah, well, they'd find their own reference set. Vinge in his fiction does a pretty good job of trying to postulate it in The Deepness of the Sky.

As far as encounter, the rest's just pattern-matching, and finding a common language. I'm pretty sure there's a possible mathematical Babel effect too.

[bigpooch]

If you haven't looked at the philosophy of mathematics, a
good summary is at:

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

To me, it seems "heretical" to think that numbers do not
exist, especially if anyone has studied pure mathematics to
any extent! Most professional mathematicians are really
"discovering" properties of abstract entities which "exist"
in an abstract "gedankenvelt". Of course, then there is a
question of what do you mean exactly by "number" and "exist"
in the question.

Ask any mathematician if he can conceive of ANY cosmos (not
just the one we are presently in) for which the truths of
elementary number theory (such as the sum of the cubes of
two positive integers cannot be a cube of a positive integer
to use David Sklansky's example) are not the same as the
truths of elementary number theory in OUR cosmos. It is
analogous to a philosopher asking whether or not the
concept of "concept" would be completely different in a
different cosmos. [That is why God cannot find a triplet of
positive integers x,y and z for which x^3 + y^3 = z^3
(because it's impossible so nobody can find such x,y and z)
in the same way that God cannot make an absolute truth
(such as "there exists something, i.e., not nothing at all"
or a tautology such as "a statement cannot be both true and
false at the same time") a false statement.

Of course, with respect to the nonnegative integers, there
shouldn't be much of a question as to their existence as one
person replied that the positive integers can be generated
from set theory. How do these positive integers "exist"? I
think everyone has a capacity to conceive of the part of the
"gedankenvelt" where they do exist. Number theorists would
have better "vision" than someone who is just counting
blocks in preschool; nevertheless, even a blind person can
count.

Then, eventually, as you have learned from real analysis
(I assume you have studied this from your posts), real
numbers do exist as well. For example, if there were any
intelligent aliens, they would independently find e and pi
since they are such common mathematical constants (of
course, their "base" may be different from base 10, but
that's just a different way of representing the same
concept, just as words of different languages may represent
the same concept) and since these are transcendental, the
real numbers would "exist".

Now the next leap of faith is "Does i = (square root of -1)
exist?". Well, it's pretty evident that eventually, any
group of intelligent beings would see that the complex plane
is important and they would eventually discover the
relation e^(i*x) = cos (x) + i sin (x).

Of course, any group of intelligent aliens may also follow
the same course as humans:

a) find x where x+5 = 2 (negative numbers)
b) find x where x*5 = 2 (rational numbers)
c) find x where x*x = 2 (algebraic numbers)
d) find pi
e) find x where x*x = -1

For the more "exotic" numbers, they exist, but most people
don't have a clear picture of them. For anyone in pure
mathematics, complex numbers do in a sense exist, and they
are "complete" in an algebraic sense.

Now, numbers don't just exist as a set of disparate entities
but rather as a set of objects with some operations, so as
you almost certainly know, the kinds of algebraic objects
that are often studied are groups, fields and rings. I
would not conceive of any group of intelligent beings that
are capable of interstellar travel that would not already
have a developed theory of abstract algebra since you
encounter algebraic objects often in physics.

Then, of course, there are more abstract entities such as
sets, functions, tensors, yada yada yada....
and different "velts" such as Euclidean n-space, Riemann
manifolds, yada yada yada

[chezlaw]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Yes they do. Ask them. Almost all mathmeticians count theselves as Platonists or Formalists or occasionaly Constuctionists. But they don't talk about it much.

[/ QUOTE ]
They really don't. The question is not a mathematical one and most have no interested in hi falootin stuff.


[/ QUOTE ]
Although it may be true that they dont argue about it actively much, I expect this is because they consider it unresolvable. Every mathematician in my department was at least familiar with the 3 different positions DS refers to and all had some opinion, no matter how half-hearted.

[/ QUOTE ]
The ones I thinking of are profundly disintrested in anything philosophical (anything much really except maths, girls, curry and beer).

I have no doubt that numbers exist but I don't know what it means for them to exist.

chez

[luckyme]

[ QUOTE ]
Would a super-intelligent race of insectoid aliens have the same conception of numbers as humans? Maybe, maybe not. If they were highly predisposed to hive-mind thinking, they might be unable to distinguish single entities and instead see things as part of larger groups.

I know that example is kind of weak, but I do think there's an inevitable prejudice towards thinking that our modern human conception of numbers is the only possible conception.

[/ QUOTE ]

It may be weak, but the argument that an intelligent species needs our numbers of various types should be challenged. Beaver dams, bee hives are some that come to mind. What concept of numbers do they work in to create a honeycomb for example. Does an ape throwing a rock need a course in ballistics ( oops that's me)?
I plead mathematical ignorance but I'm skeptical of things like the inverse square law being the only way to frame that issue, for example. If it's not, then why would we consider it a part of nature, of existing ( other than in the obvious 'I can think of it therefore it exists' view, which of course requires no discussion).

luckyme

[carlo]

[ QUOTE ]
I mean in the platonic sense - separate from the physical universe, eternal, timeless, unchanging, etc etc. Or are they purely constructs of language, relationships between physical things, something else?

I'd be interested in people's opinions - it seems the only thing I still find mysterious

[/ QUOTE ]

The individual numbers,1,2,3.. are an abstraction devoid of life. An approach to "number" would better be related to taking a ball of putty or a whole loaf of bread. Splitting the whole into separate parts will give you the experience of the number "2". Likewise splitting a part again will give the experience of the number "3". In this procedure you will obtain a feeling of the process and the consequent result. Looking at this one could perceive that "number" is a analytic(catqbolic,destructive,?) process in which the whole is metamorphosed into parts.

In the present number system the process is an operator of "the one" and in this presents it's abstraction. There is no connection of the previous to the later save by another abstraction.

The experience of this activity will offer a thought, feeling, within a will force which brings this to perception. Abstracted, we only see "1" and the next is "2" which does not contain any of the "1".Only abstract operations attempt to bring these together but the reality lies within the individual perception which when combined with the concept completes the reality.

Number will not be found in the external world but it and mathematics arise within the human soul from the most hidden which are the "will forces".

To furthur clarify, we are most in tune with our "thoughts". There is nothing clearer to the human being than his thoughts and life proves this. In our feelings we are not so clear. We could perhaps come to the conclusion that "we aren't so smart" and in this use our thinking but appreciation of feeling is not so clear. Attempting to conme to grips with our feeling doesn't readily lend itself to "thoughts" and in this we have a great plethora of human activities, both benign and malevolent. The least and most hidden from us is our "will". If one picks up his arm the thought comes to mind and the arm moves with an expressive feeling. The copnnection between the thought and the movement is lost and one could say totally hidden. It is that part of us which is the greater mystery.

Mathematics comes from this part of the soul and has an antecedant history which is another story. Geometry is learned by the movement of human limbs and other willful activity and thus we have Euclid. So the movements(read will) and the human perception of this as it "bubbles to the surface" can be called geometry.

Number and mathematics is real whose source is within and not without the human being.

[CityFan]

[ QUOTE ]
[ QUOTE ]
I mean in the platonic sense - separate from the physical universe, eternal, timeless, unchanging, etc etc. Or are they purely constructs of language, relationships between physical things, something else?

I'd be interested in people's opinions - it seems the only thing I still find mysterious

[/ QUOTE ]

The individual numbers,1,2,3.. are an abstraction devoid of life. An approach to "number" would better be related to taking a ball of putty or a whole loaf of bread. Splitting the whole into separate parts will give you the experience of the number "2". Likewise splitting a part again will give the experience of the number "3". In this procedure you will obtain a feeling of the process and the consequent result. Looking at this one could perceive that "number" is a analytic(catqbolic,destructive,?) process in which the whole is metamorphosed into parts.

In the present number system the process is an operator of "the one" and in this presents it's abstraction. There is no connection of the previous to the later save by another abstraction.

The experience of this activity will offer a thought, feeling, within a will force which brings this to perception. Abstracted, we only see "1" and the next is "2" which does not contain any of the "1".Only abstract operations attempt to bring these together but the reality lies within the individual perception which when combined with the concept completes the reality.

Number will not be found in the external world but it and mathematics arise within the human soul from the most hidden which are the "will forces".

To furthur clarify, we are most in tune with our "thoughts". There is nothing clearer to the human being than his thoughts and life proves this. In our feelings we are not so clear. We could perhaps come to the conclusion that "we aren't so smart" and in this use our thinking but appreciation of feeling is not so clear. Attempting to conme to grips with our feeling doesn't readily lend itself to "thoughts" and in this we have a great plethora of human activities, both benign and malevolent. The least and most hidden from us is our "will". If one picks up his arm the thought comes to mind and the arm moves with an expressive feeling. The copnnection between the thought and the movement is lost and one could say totally hidden. It is that part of us which is the greater mystery.

Mathematics comes from this part of the soul and has an antecedant history which can carry us far off. Geometry is learned by the movement of human limbs and other willful activity and thus we have Euclid. So the movements(read will) and the human perception of this as it "bubbles to the surface" can be called geometry.

Number and mathematics is real whose source is within and not without the human being.

[/ QUOTE ]

The word I'm looking for is GIBBERISH

[bunny]

[ QUOTE ]
It may be weak, but the argument that an intelligent species needs our numbers of various types should be challenged. Beaver dams, bee hives are some that come to mind. What concept of numbers do they work in to create a honeycomb for example. Does an ape throwing a rock need a course in ballistics ( oops that's me)?
I plead mathematical ignorance but I'm skeptical of things like the inverse square law being the only way to frame that issue, for example. If it's not, then why would we consider it a part of nature, of existing ( other than in the obvious 'I can think of it therefore it exists' view, which of course requires no discussion).

luckyme

[/ QUOTE ]
It's clearly not essential that they discover numbers - we dont solve differential equations when we catch a ball. Nonetheless, if they choose to examine the properties of functions, there are some that are differentiable and some that arent - there is no choice about that, merely about whether we ask the question. In your inverse square law example - any species looking into it must discover that relation imo - irrespective of culture (a simpler example maybe is pi - the ratio between the circumference and diameter of a circle, even though "real" circles dont exist in nature, all cultures that have noticed it as a constant agree on its value).

It's how it seems to me and pretty much any mathematician I know, anyhow - but I agree it's something worth challenging.

[Mickey Brausch]

[ QUOTE ]
Why should the numbers go away when we do (which also seems to contradict the timeless, eternal comment you made)? I cant see why they are dependant on us - Pythagoras' theorem was true before anyone thought of it.

[/ QUOTE ]I'm saying there will be nobody around to call them "numbers" and "Pythagoras' theorem". Of course, the physical laws as such will not change because Man will no longer be there.

Mickey Brausch

[luckyme]

[ QUOTE ]
In your inverse square law example - any species looking into it must discover that relation imo - irrespective of culture (a simpler example maybe is pi - the ratio between the circumference and diameter of a circle, even though "real" circles dont exist in nature, all cultures that have noticed it as a constant agree on its value).

[/ QUOTE ]

I was looking at several topics -
"Exist" can be used ( and often is on this forum) to include everything conceivable so doesn't really add anything to a discussion.
Could an alien species do some pretty impressive feats and not have an awareness of numbers as we know them ( several commented along the lines of "any species capable of X ".
Is it possible to express concepts like Pi in terms of, say, a spectrum where the midpoint is the equivalent of our 1/1 and Pi would sit at light orange. I don't see how we can be sure that there isn't other systems non-number based that express relationships and match up with what we find out there. Is there some proof of that?

luckyme

[bunny]

[ QUOTE ]
I was looking at several topics -
"Exist" can be used ( and often is on this forum) to include everything conceivable so doesn't really add anything to a discussion.

[/ QUOTE ]
This certainly seems to be the crux of my question - do things which we only experience through conceiving them (pi, square root of minus 1, etc) exist if nobody has actually got around to conceiving them yet, or ever will? It seems to me that they do - but this makes our knowledge of them mysterious.

[ QUOTE ]
Could an alien species do some pretty impressive feats and not have an awareness of numbers as we know them ( several commented along the lines of "any species capable of X ".

[/ QUOTE ]
I think it certainly is possible, my question was perhaps poorly phrased in that I meant abstract mathematical objects in general, rather than specific numbers.

[ QUOTE ]
Is it possible to express concepts like Pi in terms of, say, a spectrum where the midpoint is the equivalent of our 1/1 and Pi would sit at light orange. I don't see how we can be sure that there isn't other systems non-number based that express relationships and match up with what we find out there. Is there some proof of that?

luckyme

[/ QUOTE ]
I'm not sure how this is different - my contention would be that that would constitute just another language - the point is the same theorems would be true (the area of a circle would be half pi times the radius squared - however that concept would be translated into the light orange language.

Another way of saying this - if we could translate their light orange mathematical statements, we could derive new results, translate them back and the aliens would agree that we had produced a new theorem.

[luckyme]

[ QUOTE ]
I'm not sure how this is different - my contention would be that that would constitute just another language - the point is the same theorems would be true (the area of a circle would be half pi times the radius squared - however that concept would be translated into the light orange language.

Another way of saying this - if we could translate their light orange mathematical statements, we could derive new results, translate them back and the aliens would agree that we had produced a new theorem.

[/ QUOTE ]

I'll try a couple - we see a lot of human faces when we look at things, would an alien species see human faces in a knotty pine board. No. So is the human face a property of the board or of my mind.

We study nature we see numbers, but we are looking with numbers,we aren't going to see it in terms of how it sounds to a wasp from Atares.

Does nature have properties that don't translate into our mathematics? How would we know? Perhaps some of the weirdness of QM hints at that, for example.

It does seem like mathematics snuggles right up to reality, I'm just suspicious because I see a lot of knotty faces.

luckyme

[FortunaMaximus]

You know, one of the things I always found interesting about the Uncertainity Principle was that you could harness a finite set of particles and regenerate them several galaxies away. So random Adam and Eve's disappear, and hell, they could even be named Adam and Eve, and they would end up on a planet with the perfect biosphere...

The Universe is big enough for that, and the process would just be weird enough to happen everyday. Who knows. Wherever you go, there you are.

Oh, well. "Go now, for there are worlds other than those."

[bunny]

[ QUOTE ]
I'll try a couple - we see a lot of human faces when we look at things, would an alien species see human faces in a knotty pine board. No. So is the human face a property of the board or of my mind.

[/ QUOTE ]
I'd say it's your mind.

[ QUOTE ]
We study nature we see numbers, but we are looking with numbers,we aren't going to see it in terms of how it sounds to a wasp from Atares.

[/ QUOTE ]
I would contend that if we could translate from antaran wasp-speak to maths, we could understand what they had discovered about nature, make deductions and translate it back into something their scientists would understand and agree with.

[ QUOTE ]
Does nature have properties that don't translate into our mathematics? How would we know? Perhaps some of the weirdness of QM hints at that, for example.

[/ QUOTE ]
This is a fascinating idea. I'd never thought of the possibility before. It's always just seemed to me that it's a hint that we havent got it right yet.

[Mickey Brausch]

[ QUOTE ]
[ QUOTE ]
They really don't. The question is not a mathematical one and most have no interested in hi falootin stuff.

[/ QUOTE ]

They used to. But I hope you are right.

[/ QUOTE ]I hope you are wrong.

[chezlaw]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
They really don't. The question is not a mathematical one and most have no interested in hi falootin stuff.

[/ QUOTE ]

They used to. But I hope you are right.

[/ QUOTE ]I hope you are wrong.

[/ QUOTE ]
There's hope for you both, I reckon most never used to.

chez

[madnak]

[ QUOTE ]
This certainly seems to be the crux of my question - do things which we only experience through conceiving them (pi, square root of minus 1, etc) exist if nobody has actually got around to conceiving them yet, or ever will? It seems to me that they do - but this makes our knowledge of them mysterious.

[/ QUOTE ]

They definitely don't exist in a certain form. For example, the Greek letter pi, or the decimal sequence 3.14159..., didn't exist before humanity.

But whenever a circle has existed, its circumference has had a certain ratio to its diameter. This is just as much a meaningful representation of pi as any of our symbols - thus nature itself "wrote out" pi many times before human being ever existed. Just like the light-orange expresses pi in one language, a circle itself expresses pi in another language.

Why does that ratio appear so frequently in our universe? Is it some property of our universe itself? Are there universes in which circles can't exist, or in which the ratio of a circumference of a circle to its diameter isn't pi? The fact is, something in the fabric of our universe indicates pi somehow. If we knew exactly what and how, the problem might be trivial.

[madnak]

[ QUOTE ]
I'll try a couple - we see a lot of human faces when we look at things, would an alien species see human faces in a knotty pine board. No. So is the human face a property of the board or of my mind.

[/ QUOTE ]

Both. The shape of the board is a property of the board, your interpretation of that shape as resembling a human face is a property of your mind. This really isn't analogous to mathematical principles. We've used many means external to our minds to validate the fact that the ratio of the circumference of a circle to its diameter is 3.14159... We can do no such thing with the board.

[ QUOTE ]
We study nature we see numbers, but we are looking with numbers,we aren't going to see it in terms of how it sounds to a wasp from Atares.

[/ QUOTE ]

Maybe not, but when we send a shuttle to the moon, the shuttle reaches the moon. A wasp from Atares might interpret it very differently, but it would still recognize in some form that we "got it right," that our shuttle came into contact with the moon.

[ QUOTE ]
Does nature have properties that don't translate into our mathematics? How would we know? Perhaps some of the weirdness of QM hints at that, for example.

[/ QUOTE ]

To me, it seems arrogant to assume otherwise. Have human beings, a very recent development at best, reached the point of being able to understand every facet of everything? I'm almost certain there are "intellectual" faculties which we haven't developed. I'm sure if we were to create an AI (based on heuristics, say, rather than modeling of the human brain) then that AI would quickly reach levels of understanding that would make humans look like donkeys or eels.

BUT that's not an indication that there's no basis to the things we do understand.

[bunny]

[ QUOTE ]
[ QUOTE ]
This certainly seems to be the crux of my question - do things which we only experience through conceiving them (pi, square root of minus 1, etc) exist if nobody has actually got around to conceiving them yet, or ever will? It seems to me that they do - but this makes our knowledge of them mysterious.

[/ QUOTE ]

They definitely don't exist in a certain form. For example, the Greek letter pi, or the decimal sequence 3.14159..., didn't exist before humanity.

[/ QUOTE ]
No this seems the easy bit.

[ QUOTE ]
But whenever a circle has existed, its circumference has had a certain ratio to its diameter. This is just as much a meaningful representation of pi as any of our symbols - thus nature itself "wrote out" pi many times before human being ever existed. Just like the light-orange expresses pi in one language, a circle itself expresses pi in another language.

Why does that ratio appear so frequently in our universe? Is it some property of our universe itself? Are there universes in which circles can't exist, or in which the ratio of a circumference of a circle to its diameter isn't pi? The fact is, something in the fabric of our universe indicates pi somehow. If we knew exactly what and how, the problem might be trivial.

[/ QUOTE ]
What I find puzzling is that, in fact, circles dont exist in the physical world (not real, perfect ones anyhow) - pi is an abstraction. I agree with you that a perfect circle would be just as good a representation of pi, but not that my "circular" table is (for example).

[madnak]

Good point, and interesting. Pi may not actually exist, I suppose. I don't know enough about the true geometry of the universe to say. If it's 11-dimensional, and non-Euclidean, then who knows which of our mathematical concepts have "true" universal analogues?

But it seems that there must be some mathematical "realities," even if they aren't the ones we talk about.

[bunny]

[ QUOTE ]
Good point, and interesting. Pi may not actually exist, I suppose. I don't know enough about the true geometry of the universe to say. If it's 11-dimensional, and non-Euclidean, then who knows which of our mathematical concepts have "true" universal analogues?

But it seems that there must be some mathematical "realities," even if they aren't the ones we talk about.

[/ QUOTE ]
My view is that they exist anyway - even if there is no instantiation of the concept within the physical world (how else do you explain chinese, europeans and central americans all "discovering" pi - with no actualisation of it in reality?).

What I find mysterious is that a physical thing can interact with these non-physical entities. It leaves a niggling doubt that perhaps there is some physicalist account for mathematical entities but it goes against the grain.

[madnak]

[ QUOTE ]
My view is that they exist anyway - even if there is no instantiation of the concept within the physical world (how else do you explain chinese, europeans and central americans all "discovering" pi - with no actualisation of it in reality?).

[/ QUOTE ]

Pi may, like Newton's Laws of Motion, be a good approximation for how things work "around here." I do tend to agree with you that there's some kind of inherent reality to basic mathematical principles, although I think we humans can have no idea what such a reality "looks like."

[ QUOTE ]
What I find mysterious is that a physical thing can interact with these non-physical entities. It leaves a niggling doubt that perhaps there is some physicalist account for mathematical entities but it goes against the grain.

[/ QUOTE ]

A physical thing can identify patterns and then make predictions based on those patterns. One of the things humans are very good at. We don't need to contact a specific physical element to identify a pattern - we may need to create some kind of analogue to the pattern inside our brain, but I'm not sure if that's here or there.

Incidentally, I was doing a google search for something on 2p2, and I found an article (http://www.sciencedaily.com/releases/2006/04/060425015333.htm) that's interesting and maybe even relevant.

[thylacine]

[ QUOTE ]
I mean in the platonic sense - separate from the physical universe, eternal, timeless, unchanging, etc etc. Or are they purely constructs of language, relationships between physical things, something else?

I'd be interested in people's opinions - it seems the only thing I still find mysterious.

[/ QUOTE ]

The answer to an existence question is always implied by the answer to an enumeration (counting) question. So the question to ask is:

HOW MANY NUMBERS EXIST?

[bigpooch]

So, if your view is that numbers exist, there are other
questions and thoughts that immediately follow.

I'll share with you my view, which you may agree or disagree
with, so that perhaps you may see another "piece of the
puzzle".

Numbers are concepts and obviously some concepts "exist",
although by what manner they exist, you may have your own
view of how they exist. Clearly, some concepts men think
of don't really exist but many concepts (those for which
almost every language has a "signifier") do. Also, just
because there is a "signifier" does not imply there is
truly a "signified" that really exists!

Those concepts dealing with the physical cosmos, such as
time and space belong to the realm of physics and the manner
which these exist obviously depend on the laws of physics.
The faith of the "priests of science" rest heavily on
empiricism, but don't let that lead you to positivism!

On the other hand, there are concepts of philosophy which
seem to exist in any cosmos regardless of the properties of
that cosmos because for any such cosmos, there must be a
"core" cosmos of these concepts. What's in this "core"
other than numbers?

I would think that "truth", "existence" (in a platonic
sense, as you mentioned), "concept" (something that exists
that can be conceived by something that can conceive) are
among some of these. A clearer or more precise
understanding of these fundamental ideas would greatly help
in understanding more complex ideas.

Then, there are more complex ideas that depend on the
existence of "beings with free will" (a good question is
"Would they necessarily exist?") such as "good", "free", and
"justice".

Of course, that leads to one of the most controversial ideas
of all (if not the most important, in the eyes of many),
namely, God. Theologians, philosophers and others can
quibble about the attributes of God ad infinitum ad nauseum.

First of all, it's impossible to prove the existence of
God in the same manner as a mathematical or philosophical
proof. On the other hand, belief in God is rational as long
as you believe in the existence of "other minds": see
Plantinga's book "God and Other Minds" where he shows the
fallacy of the proofs and the rationality of belief in God.
Not everyone will agree with the book, but I believe that
the main thesis of the book is correct, i.e., there does
not exist any proof of God's existence that is sufficient
for the philosopher (no need to ask the empiricist, as they
are "blind" to even the existence of certain classes of
concepts!), but belief in God is rational (it's equivalent
to the belief about other minds).

Where does that leave us?

It's clear that faith is necessary and Sklansky is
essentially correct about the necessity of faith.

Of course, this leads to questions about religious beliefs
and I do share with you a very strong sense that the
Abrahamic monontheists are essentially on the right track
and that the closest track to the truth, but the one with
many ( and I mean several!) problematic doctrines is
Christianity. The Reformation tried to put things back on
track, but unfortunately, some beliefs, such as biblical
inerrancy were taken to such an extreme to create groups
such as fundamentalists. Also, if you have read much of the
New Testament, you can make your own judgment about Roman
Catholicism!

Now, Leibniz was a very intelligent man, and gave his
thoughts about "monads" in his worldview, which I hesitate
to accept as "correct", but would not be surprised if it
were. Also, although I haven't read much of Swedenborg,
I believe he was deceived or deluded. Both of these two
men were highly intelligent and it's very unlikely that any
2+2er will be smarter than either one of them. Of course,
being intelligent is no immunity from being badly mistaken.
Nevertheless, I trust Leibniz. Of the theologians that I
trust, I trust Karl Barth the most, although I would also
like to read more about Paul Tillich's thoughts.

Also, a caveat: "Knowledge puffs up...". Need I quote
chapter and verse?

[CityFan]

[ QUOTE ]
Good point, and interesting. Pi may not actually exist, I suppose. I don't know enough about the true geometry of the universe to say. If it's 11-dimensional, and non-Euclidean, then who knows which of our mathematical concepts have "true" universal analogues?

But it seems that there must be some mathematical "realities," even if they aren't the ones we talk about.

[/ QUOTE ]

Without a doubt, if the natural numbers (1,2,3...) exist then pi also exists, and has the value that it has. It exists as a property of the abstract "circle", which exists in an abstract flat plane, and has nothing to do with the actual physical properties of our universe.

From a physical point of view, any circle that you care to "draw" will not have a diameter exactly equal to 2pi times its radius - and I'm not talking about wobbling the pencil here!

Relativity tells us that space-time is curved. Space may be approximately flat in the area in which you draw your circle, but not completely flat. It may be positively curved, so that the circle actually has a cirmumference less than 2pi x radius (imagine drawing a circle on the surface of a sphere, and measuring the "radius" along the surface), or it may be negatively curved so that the circle has a cirmumference greater than 2pi x radius (harder to conceive, but equally possible).

Depending on the curvature of space, any circle in the physical universe will have a circumference/radius ratio which is slightly different from pi. Pi only exists in the abstract flat plane.


*In fact the whole idea of drawing a cicle in 3-D space assumes that you can pick some kind of 2-D subspace in which to draw it. The argument above is more watertight, but essentially the same, if you replace the circle with a sphere and the circumference with either surface area or volume.

[Mickey Brausch]

[ QUOTE ]
The answer to an existence question is always implied by the answer to an enumeration (counting) question. So the question to ask is:

HOW MANY NUMBERS EXIST?

[/ QUOTE ]Depends on who and how is doing the numbering. If a numbering is done.

How many trees are falling in the forest when no one hears them?

Mickey Brausch

[FortunaMaximus]

A vast tree, liquid-crystalline in structure, ever-expanding. White-bluish-purplish in its structural outside, the lifeblood of numbers within, filling the crevices and ever-expanding concurrently.

So the Gordelians tell me. And I tend to take 145-billion-year old entities at their word, when they describe infinity and the true nature of the multiverse.

~FM

[chezlaw]

/handwaving

how about the concept of numbers is an attractor in concept space. Then the concept of numbers exists awaiting the existence of creatures capable of concepts.

providing that concept space exists independently of concept capable creatures.

chez

[CityFan]

Nah,

If concept space exists then every point in concept space exists in potentia, whether it be an attractor or not. So the concept of numbers being an attractor is irrelevant with regards their existence.

I'd like to know a bit more about this concept space though.

[wazz]

Oh man, this is so unfair, I scrolled through about a quarter of this thread thinking 'there's so much I have to say!' then I see it's all been said already.

I think though there's a need for some sort of more rigorous definition of 'exist' and 'numbers'. I also think it is interesting to ask whether it's possible to conceive of a world without numbers - I would tend to believe, yes, a world where there is something of everything, no more, no less, and unquantifiable. No relationship or association between things - numbers are meaningless in such a world.

In our world - I would say that, as we understand it, numbers do exist independent of any rational beings, whether or not there are rational beings to understand how these numbers affect the physical world, they do affect it. This is analagous to saying that, if a circle existed independent of a rational mind, then it would follow that the very presence of an exact relationship between ratio and circumference as a constant number means that number exists independently of our rational minds. Whether or not there is a rational mind to discover that ratio is not important. The golden ratio is also a good example.

If you extend the question further - do imaginary numbers exist in the classical sense? If so, would they exist independently of the rational beings who imagine them?

[chezlaw]

[ QUOTE ]
Nah,

If concept space exists then every point in concept space exists in potentia, whether it be an attractor or not. So the concept of numbers being an attractor is irrelevant with regards their existence.

I'd like to know a bit more about this concept space though.

[/ QUOTE ]
True, I'm up for that but its the attractors that have the special quality of not being arbitary.

chez

[CityFan]

For me it's all or nothing. Either the whole of mathematics "exists" or none of it does. So if the natural numbers exist, then imaginary numbers definitely exist, as do transcendental numbers and nowhere-differentiable continuous functions.

[FortunaMaximus]

[ QUOTE ]
Nah,

If concept space exists then every point in concept space exists in potentia, whether it be an attractor or not. So the concept of numbers being an attractor is irrelevant with regards their existence.

I'd like to know a bit more about this concept space though.

[/ QUOTE ]

Of course. It's best to think of an infinite set of numbers as a seed. An acorn, if you will.

It needs consciousness and will to nourish its growth. Dead matter, water, what have you. The point of this Universe is to fill every bit of matter with consciousness. A big bang to expand the coordinate points and disperse the matter over a vast space.

c is a necessary speed limit, because if the pace of growth is too fast, it will crash and fold into itself. And retrigger its initial starting conditions, but with a random tweak of the fundamental forces. There have been countless universes before this. And simply put, they were mistakes. But the next Universe in the manifold eats these mistakes and they are emergent in this and the next universe.

It may take googolplexes of incarnations before there's a stable, sustainable m-space. (manifold space)

So God's just a tool for a primitive civilization, and to stem its growth just so. Too many civilizations go extinct when they take the tools upon themselves to tinker with matter. Stars go boom. That has consequences for not only the civilization, but the surronding potential space. The irony is, though, that those novas actually create denser matter, which is in and of itself beneficial to life.

Hmm. What of the lives lost? The individuality? In the whole structure, they regenerate and finally get it right. But they cannot be interfered with.

Hope that helps. It's just a datapacket.

~FM (Echoer of civilizations)

[wazz]

[ QUOTE ]
Of course. It's best to think of an infinite set of numbers as a seed. An acorn, if you will.

It needs consciousness and will to nourish its growth. Dead matter, water, what have you. The point of this Universe is to fill every bit of matter with consciousness. A big bang to expand the coordinate points and disperse the matter over a vast space.

c is a necessary speed limit, because if the pace of growth is too fast, it will crash and fold into itself. And retrigger its initial starting conditions, but with a random tweak of the fundamental forces. There have been countless universes before this. And simply put, they were mistakes. But the next Universe in the manifold eats these mistakes and they are emergent in this and the next universe.

It may take googolplexes of incarnations before there's a stable, sustainable m-space. (manifold space)

So God's just a tool for a primitive civilization, and to stem its growth just so. Too many civilizations go extinct when they take the tools upon themselves to tinker with matter. Stars go boom. That has consequences for not only the civilization, but the surronding potential space. The irony is, though, that those novas actually create denser matter, which is in and of itself beneficial to life.

Hmm. What of the lives lost? The individuality? In the whole structure, they regenerate and finally get it right. But they cannot be interfered with.

Hope that helps. It's just a datapacket.

~FM (Echoer of civilizations)

[/ QUOTE ]

I'll have 2 of what you're having please.

[FortunaMaximus]

What, acorns? Why not. &lt;tosses 'em onto the table&gt;

[Philo]

[quote

The answer to an existence question is always implied by the answer to an enumeration (counting) question. So the question to ask is:

HOW MANY NUMBERS EXIST?

[/ QUOTE ]

That's easy. All of them.

[bigpooch]

So, you're a "full-blooded platonist"? Join the crowd! A
good explanatory link is at:

http://plato.stanford.edu/entries/platonism/

[CityFan]

[ QUOTE ]
So, you're a "full-blooded platonist"? Join the crowd! A
good explanatory link is at:

http://plato.stanford.edu/entries/platonism/

[/ QUOTE ]

Apparently so!

Although, does this mean that a rational square root of two exists?

[bigpooch]

In reality, no. You imagine that it exists and then, you
see why it's just a figment of your imagination (of course,
you know at least one proof!).

[thylacine]

[ QUOTE ]
[ QUOTE ]


The answer to an existence question is always implied by the answer to an enumeration (counting) question. So the question to ask is:

HOW MANY NUMBERS EXIST?

[/ QUOTE ]

That's easy. All of them.

[/ QUOTE ]

Not so fast, my friend. The statement "all numbers exist" does NOT imply the statement "numbers exist". /images/graemlins/smirk.gif

[Philo]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]


The answer to an existence question is always implied by the answer to an enumeration (counting) question. So the question to ask is:

HOW MANY NUMBERS EXIST?

[/ QUOTE ]

That's easy. All of them.

[/ QUOTE ]

Not so fast, my friend. The statement "all numbers exist" does NOT imply the statement "numbers exist". /images/graemlins/smirk.gif

[/ QUOTE ]

This is true. Btw, I also think that numbers exist.

The 'counting' issue that is normally asscoiated with existence questions is not "how many exist" but "what are the criteria of existence for entities of that kind?" The famous phrase from Quine is "No entity without identity," which means that we should not countenance the existence of any entity for which we do not have criteria of identity (and hence, criteria for individuating entities of that kind, which allows us to be able to 'count' entities of that kind).

Fwiw, I'm a non-criterialist, which means that I think there are no 'criteria of identity', and thus I think Quine was wrong.

[thylacine]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]


The answer to an existence question is always implied by the answer to an enumeration (counting) question. So the question to ask is:

HOW MANY NUMBERS EXIST?

[/ QUOTE ]

That's easy. All of them.

[/ QUOTE ]


Not so fast, my friend. The statement "all numbers exist" does NOT imply the statement "numbers exist". /images/graemlins/smirk.gif

[/ QUOTE ]

This is true. Btw, I also think that numbers exist.

The 'counting' issue that is normally asscoiated with existence questions is not "how many exist" but "what are the criteria of existence for entities of that kind?" The famous phrase from Quine is "No entity without identity," which means that we should not countenance the existence of any entity for which we do not have criteria of identity (and hence, criteria for individuating entities of that kind, which allows us to be able to 'count' entities of that kind).

Fwiw, I'm a non-criterialist, which means that I think there are no 'criteria of identity', and thus I think Quine was wrong.

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Just checking ... you are basically bring up the question, `whats does "=" mean?', right?

[Philo]

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The answer to an existence question is always implied by the answer to an enumeration (counting) question. So the question to ask is:

HOW MANY NUMBERS EXIST?

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That's easy. All of them.

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Not so fast, my friend. The statement "all numbers exist" does NOT imply the statement "numbers exist". /images/graemlins/smirk.gif

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This is true. Btw, I also think that numbers exist.

The 'counting' issue that is normally asscoiated with existence questions is not "how many exist" but "what are the criteria of existence for entities of that kind?" The famous phrase from Quine is "No entity without identity," which means that we should not countenance the existence of any entity for which we do not have criteria of identity (and hence, criteria for individuating entities of that kind, which allows us to be able to 'count' entities of that kind).

Fwiw, I'm a non-criterialist, which means that I think there are no 'criteria of identity', and thus I think Quine was wrong.

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Just checking ... you are basically bring up the question, `whats does "=" mean?', right?

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Well, not quite. We know what "=" means if we are using it to represent the identity relation.

The relation of identity, or 'strict numerical identity' as it is sometimes called, means this: If a and b are numerically identical then they are one and the same thing (so, a=b).

Mark Twain and Samuel Clemens are numerically identical, for example, since they are one and the same person. The morning star and the evening star (the planet Venus) are also numerically identical.