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Language, thought, and infinty

 
  

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We're The Great Old Ones Now
09:21 / 03.04.03
What does 'infinite' mean?

What is an infinite amount of space?

The human mind steps back from it. We take refuge in the illusion of motion, going away from something indefinitely. We live relatively; infinity is not relative.

In the Fregean use of Sense and Reference (where Sense means an object X as it is described, and Reference means the actual thing behind the language, to which the term refers - so "Clark Kent" has the sense "mild mannered reporter for the daily planet" and all that the Kent persona entails, but its reference is the dark-haired refugee from Krypton who is both Superman and Kent) 'infinty' seems to have a limited sense - we cannot fully express the thing in our minds, we use a kind of conceptual shorthand - and a reference we cannot hope to grasp.

Infinity crops up everywhere - God is reputedly infinite - in which case almost anything we say about him is meaningless. The argument about absolute truth in this forum is also a part of this.

We have words for things for which we have only the barest of concepts. The word seems to give a shorthand of the concept, which in turn indicates the reality.

So, two things:

How badly do we go wrong when we talk imponderables like this?

How does this business of part-formed concepts work? We seem to use words to shield our minds from ideas whose consideration ends in a regress. They're like equations - if you actually run the instruction, you get infinity, but if you just use the terms, you can manipulate them without getting too badly burned...
 
 
Sax
10:26 / 03.04.03
I think the concept of infinity is kind of a circuit-breaker for the mind... if you had to imagine "the infinity of outer space" you'd probably end up just staring into the middle distance until they carted you away. But to quantify sheer endlessness as the word "infinity", by which we just mean "a really big box, the biggest one I can imagine, with everything in it and, erm, I'd rather not think about what's outside it" we bring the shields down and have some kind of conceptual idea about what we're talking about without going too deeply into it.

I reckon.
 
 
Loomis
10:57 / 03.04.03
We need a concept of infinity to define ourselves against. If there was no concept of infinity, what would it mean to say that something was finite? And if nothing is finite, then where do you go from there? I think the term infinity doesn't say as much about infinity itself (if there is such a thing) as it says about everything else that is defined in opposition to it. It's a way of allowing us to talk meaningfully about our world, and be conceptually complete, without needing to worry about the reality of infinity.

For instance, you can define parallel lines as 2 lines which never meet. But then my maths teacher told me that you can define parallel lines as 2 lines which meet at infinity. It's the same thing, but the second definition reveals a lot more, without needing to tell you precisely what infinity is.
 
 
Quantum
11:34 / 03.04.03
'Infinity' as a word covers a lot of ground, like 'Love'. There are different types of infinity just as there are different types of love.

The infinite distance of space is not the same as the infinity of sequential numbers, is not the same as the infinity of Pi etc.

re: Frege, can we ever grasp a reference? 'The Morning star' and 'the Evening star' have different senses, same referent (Venus). 'Venus' has a sense, but the referent is a physical thing, not linguistic- Frege presupposes an objective link between language and reality.

The question of infinity relates to paradoxes- an infinite regress or a circular argument indicate something is wrong with the argument. Infinity is a useful fiction like irrational numbers- as Nick says "if you actually run the instruction, you get infinity, but if you just use the terms, you can manipulate them without getting too badly burned"
 
 
Lurid Archive
12:31 / 03.04.03
This is probably innappropriate but infinity has been a long studied concept in maths and has some odd implications. I think the best way to summarise what happens with a formal treatment of infinity is that our intuition for what it is goes out the window.

So one tends to go pretty badly wrong when talking about infinity, though this needn't be the case. It is just an area where loose reasoning gets rewarded with absurdities. (In fact, you always get things that seem like absurdities, just that with care you can keep to consistent absurdities.)

What is infinite space? There are a couple of answers I can think of, but I guess that Nick is asking about the meaning of an infinite distance? weeeellll. Does it have to have a meaning in terms that are acceptable to our finitely biased brains? I don't think so. Then again, many physicists seem to reject the notion of infinity on common sense grounds except as a convenient tool.

So, I'm not sure that the presence of the word implies existence. But then I spend a fair amount of time thinking and talking about things whose existence is partly a matter of taste.

Also, I'm not sure why infinity can't be relative. Surely an infinity to one can be a finitude to another? Just like the individual colour red can be an infinite variety of "reds". Depends how you look at it.


BTW - Its a pretty critical view that thinks of irrational (or even complex) numbers as a "fiction", any more than other numbers.
 
 
Quantum
12:50 / 03.04.03
I hold a pretty critical view :-) but relating it to Frege, irrational and complex numbers have a sense but no referent- you can't point to a thing in the world and say 'That's an irrational number'.
Don't get me wrong, I equate it to other useful fictions like natural laws, causation and induction. (I hold a pretty critical view!)

Mathematics is probably the best way to examine infinity (I read 'White Light' by Rudy Rucker recently, top quality and very relevant to this). There are (apparently) levels and layers, orders and magnitudes of infinity.
For example, take an infinitely large number. Square it. Cube the result. Divide that by the square root of minus one (i). All the stages are distinct, yet all infinite (or infinitesimal). So what sort of infinity are we talking about?
 
 
Quantum
12:55 / 03.04.03
"...any more than other numbers" sorry, should have pointed out that numbers are a fiction as much as beauty, superman or value. Numbers are a category we impose on the world, a logical structure that is in our mind not the world.
 
 
grant
12:58 / 03.04.03
Actually the Venus thing is interesting, since neither you nor I have been to Venus, and we really don't know much about Venus other than that it's a little light representing a big planet a long way away, and our conceptions of what that planet is actually like are probably not 100% cognate. So the referent is shifty. Non-identical for the same sense, maybe.

Which seems a bit like the infinite as the unknown....

Let me see if this is relevant: I'm teaching myself Mandarin, right, with these language CDs. In Mandarin, when you ask certain questions, (it seems like) the sentence is formed as a statement with a "blank" term where normally there'd be a value. So to ask "What time is it?" you actually say, "Now, it's blank o'clock," or just "It's blank o'clock." (Ji dian jong).
The person you're talking to then answers by filling in the blank: "It's eight o'clock." (Ba dian jong).
Because the language is tonal, having a question mark is meaningless... "dian jong" ends on higher pitch whether you're asking a question or answering one.
It's the *blank* that defines the question. It's reference-less.

Which seems like a kind of linguistic algebra, in a way.

In this case, you're meant to fill the terms in, but with concepts like "infinity," you can approximate, it seems like. You're left with a kind of sense-only, maybe.

What's the reference for "nothing" anyway?
 
 
grant
13:02 / 03.04.03
For example, take an infinitely large number. Square it. Cube the result. Divide that by the square root of minus one (i). All the stages are distinct, yet all infinite (or infinitesimal). So what sort of infinity are we talking about?

What happens if you do that with the number 1?
 
 
Lurid Archive
13:19 / 03.04.03
you can't point to a thing in the world and say 'That's an irrational number'.

Hmmm. That seems false to me. Well, kinda. Unless you are assuming all numbers are a fiction (fair enough) in which case it is rather odd to concentrate on irrational numbers. Is the length of a hypoteneuse of a right angled triangle whose other sides have length 1 (square root of two) a fiction? Perhaps.

Complex numbers are equally well realisable.

For example, take an infinitely large number. Square it. Cube the result. Divide that by the square root of minus one (i). All the stages are distinct, yet all infinite (or infinitesimal). So what sort of infinity are we talking about?

The problem here is that none of this need have meaning. Especially the bit at the end. But in any standard treatment, squaring or cubing an infinite number gives the same infinite number. Usually. If you do this to 1 you get minus the square root of minus one.
 
 
Quantum
13:45 / 03.04.03
"Actually the Venus thing is interesting...So the referent is shifty. Non-identical for the same sense, maybe." "You're left with a kind of sense-only, maybe. What's the reference for "nothing" anyway?" (grant)
My point exactly- I should admit I have a hatred for Frege. I believe words have a Sense (what we mean by them) but they aren't connected to a Referent (a thing in the world). Language is a human construct, NOT somehow tied to the external world. Words are sense-only. (as are numbers- you can't point to a thing in the world and say 'that is 5')

"What happens if you do that with the number 1?" (grant) 1 isn't an infinite number
"Unless you are assuming all numbers are a fiction (fair enough)" I am.

"But in any standard treatment, squaring or cubing an infinite number gives the same infinite number. Usually. (Lurid Archive)
Which is counterintuitive. If I have a line of infinite length, and I place another infinite line at right angles to it, I have an infinite area. An area is 'bigger' than a line, so an infinite area is 'bigger' than an infinite line. An infinite volume is bigger than an infinite area, so either our idea of infinity is paradoxical or needs expanding.
 
 
grant
14:30 / 03.04.03
Well, my point was that 1 cubed is 1, probably in much the same way as infinity cubed is infinity.
 
 
We're The Great Old Ones Now
14:41 / 03.04.03
Ah, Quantum, I dunno. I don't see a need for a direct link between language and reality in Frege - just an acknowledgement that what's there (whatever you decide "there" means) differs from what we propose and construct as being there.

The person who gets stuck with a linkage between language and reality is Bertrand Russell, with that incredibly annoying rubbish about Unicorns. Frege gets us out of that by saying "here's the word, here's the world. See? We're wrong a lot, and we can still have words and concepts for what we're wrong about, without changing what's actually there."

Now, he may be wrong about that, according to one, rather weird interpretation of quantum mechanics, but I don't see that he's maintaining any kind of metaphysical relationship between the word and the thing - I see Sense and Reference as getting us out of that.
 
 
reFLUX
17:53 / 03.04.03
if there is infinity then nothing ends, nothing begins.

sorry that's not very helpful.
 
 
eye landed
12:03 / 04.04.03
If I have a line of infinite length, and I place another infinite line at right angles to it, I have an infinite area. An area is 'bigger' than a line, so an infinite area is 'bigger' than an infinite line. An infinite volume is bigger than an infinite area, so either our idea of infinity is paradoxical or needs expanding. (Quantum)

Indeed. An object with a fractal perimiter (infinite), does not have an infinite area. I'm not a mathematician, but I know this problem is solved through the use of non-integer dimensions, i.e. a fractal line is not one-dimensional but 1.2345-dimensional. I wish I could explain this in a lucid way, but maybe someone else knows more than I.

My point being, of course, that our infinity may not be somebody else's infinity. Imagine Mr. Square of Flatland. Would a (finite) cube be infinite to him? Or just equally incomprehensible?
 
 
Quantum
13:27 / 04.04.03
Grant- '1' is a tricky subject in itself, but I see your point. Thing is, 1 is a concrete number, you can point to one thing. You can't point to an infinite amount of things. The mathematical transformation is just an example to show how infinity is not like other numbers. How about if you take infinity and divide by ininity squared? (you get one over infinity, zero) What if you take infinity and raise it to the power of infinity? (Aleph one in some systems) What if you take infinity as x and do X! (infinity factorial)? My point is that infinity is a lot like zero in some ways, but in other ways is totally different.
It's tempting to say that infinity factorial is equal to infinity etc, but that makes a mockery of common sense. Take infinity x2- If I have an infinitely long line to my right, then another to my left, I have twice as many infinite lines, yes? We have to decide either to take infinity in a common sense understanding, or mathematical. If we take it mathematically we should just post a link to a maths site explaining the types of infinity (Aleph 1, 2 etc) if we are examining a common understanding of infinity then we should accept common sense transformations of it. The two understandings are incommensurable.

Frege- From my memory of his work he posited language as an objective system. I agree with his ideas on the senses of words, but totally disagree that the referent is connected to the word in any way- how could it be? We can only know the sense, so how can we talk about the reference meaningfully? What we think is the referent is just another sense. He uses language as a bridge between our subjective reality and the external world, which I see as an invalid move.
(I know the interpretation of quantum mechanics you mean, that's another thread!)

The idea of partial dimensions is a mathematical technique which explains fractals so that they don't have an infinite boundary, surely? But anyway it goes to show there are differing types of infinity. That's my point, the word 'infinity' maps onto many concepts.
(or it has many senses if you prefer)

"My point being, of course, that our infinity may not be somebody else's infinity. Imagine Mr. Square of Flatland. Would a (finite) cube be infinite to him? Or just equally incomprehensible?" (substatique)
A cube to him would be a square. But your point is valid, our understanding of infinity is relative, subjective and culturally influenced. We should have different words for different types of infinity.
 
 
Lurid Archive
14:13 / 04.04.03
We have to decide either to take infinity in a common sense understanding, or mathematical

There isn't any common sense understanding of infinity. There are some common sense conceptions of the finite that you can try to stretch to the infinite - at which point everyone gets shocked that it doesn't work. But it is no surprise really.
 
 
01
17:12 / 04.04.03
Infinity can't be applied to something that has borders or constraints ie, line, area or cube. We're trying to meld two concepts that can't stand being in the same room together. Oil and water. So to say that an infitely squared area is less than an infitely squared cube is paradoxical and "infintely" frustrating (ho, ho) because we are imposing restrictions on a term that's nature is define the borderless. Infinity is infinity. Plain and simple. It's a concept to define the vastness of the universe, existence, the cosmos, the Sandman, whatever.
Perplexed about the big bang and the "beginning of time"? Throw it out the window. It's linear. It's quantifiable. To get all zen infinity is the gateless gate.
 
 
01
17:28 / 04.04.03
[b]AND[/b] the universe's sole existence depends on the interaction between the bordered and the borderless. The two need each other. "Infinity" needs an increasing aggregate of objects to describe. It implies motion. If I have one apple, I have one apple. Game over. If I have two or ten I have two or ten, same thing. If I have infinite amount of apples I'm always adding one, five, five thousand, one million squared, whatever. It's like the engine of the universe.
 
 
01
18:01 / 04.04.03
And to bring language into the mix is where things start to make no sense because we're trying to marry the two. The only way is to keep them separate, the word and the definition. The word (being capsulated), is an interface for the definition (being non-capsulated), and nothing more.
 
 
—| x |—
07:32 / 07.04.03
Infinity can't be applied to something that has borders or constraints…
~zerone

But I don’t think this is right. If we look at a fractal, say the Mandelbrot, then we see that there is a definite shape—there are borders—yet the border is infinite. Moreover, many fractals, which are pictures of infinity, can be constructed inside a finite space! However, I think I get what you are saying about “…the universe's sole existence depends on the interaction between the bordered and the borderless,” and would tend to agree.

Myself, well, I always found it quite beautiful that a line—of any length, a plain—of any size, and a solid—of any shape, all have the same number of points: an infinite number! And it is the same size of infinity.

An object with a fractal perimiter (infinite), does not have an infinite area.
~substatique

Yeah, and that’s kinda’ neat too! Like you, I am unsure of the math of this, but it reminds me of the calculus, and an object sometimes known as “Gabrielle’s Horn.” It’s kinda’ like a cone, but instead of having a definite end point, its end tapers off to infinity. The result is that it has an infinite surface area, and it can be asked, “how would one paint this area?” Apparently, and I suppose this is where the calculus comes in, the Horn has a finite volume, so one merely fills the Horn with paint, and provides the condition that its surface is porous—voila!—an infinite surface painted by a finite amount of paint.

if there is infinity then nothing ends, nothing begins.
~Varis 08

But this isn’t really so! Take the set of Natural Numbers: 1, 2, 3,…there is a beginning here, but no end. As well, we can take the set of all Integers and create a beginning simply by how we order them: 0, 1, -1, 2, -2,…so where it seemed that we had an infinity in two “directions” (to the left and to the right of 0), we can simply change the way we “see” the set, and mark out its beginning. Similarly, with a fractal, again, take the Mandelbrot, there is also a beginning in the sense that there is the whole structure present before, but its “depth” is endless.

What does 'infinite' mean?” and
We seem to use words to shield our minds from ideas whose consideration ends in a regress.
~Nick

I would say that any word we use “shields our minds” from an infinite regress. Words, as has been pointed out above, have a sense. Now sense can be more or less definite, but I don’t think any word, nor list of words and/or phrases, ever adequately describes the whole of a thing in the world—it seems to me that all things that are, are in themselves infinite: in an external and internal sense. The former based on relations with other objects, the latter based on internal relations amongst constituents.

It seems to me that all our words limit the reality, or perhaps better, limit our perception of the reality. As Lurid points out, our brains/minds do seem to be biased towards the finite—we are tempered and tempted by the discrete. In other words, it appears that it is more a function of the human mind which discovers/invents separate and discrete objects, and then presumes that since these objects are seen as separate and discrete that they must also be finite. So the problem with infinity seems to be this: we have a word, say ‘shovel’, and we know the sense of this word, and it happens that the sense of this word appears to adequately correspond to an object in the world—a shovel; however, the word ‘infinity’ does not seem to have this same correspondence. To me it seems that it is self-referencing, which leads us into all sorts of regress problems: Russell’s Paradox, the Set of All Sets, etc.

I think were language and infinity get really interesting is when we think of language as being finite: it has only a certain amount of letters which form only a certain number of words, which can make only a finite number of intelligible finite sentences. And even if we take away these finite limitations, we are still left with a language that can only have an infinite number of sensible strings. Now, it can be shown that there are more functions than there are sensible strings; that is, there is a larger infinity of functions than there are strings of letters to represent these functions. A function is merely a form of relation—it maps something to something else; thus, if we think about all the apparently discrete objects in the universe, and think about all the relations that these objects have to one and other, then it seems to me that there are more relations amongst objects than there are strings of letters to describe those relations. In other words, it seems to me that there could exist many more things, in the sense that a relationship can be seen as a “thing,” than we can ever hope to put words to. So language might never be able to fully and completely capture the reality.

But then again, maybe I’ve fallen off my rocker, but, according to Zeno, I’ll never hit the floor!

Free fallin’ eZ.
 
 
Quantum
11:33 / 07.04.03
"To me it seems that it is self-referencing, which leads us into all sorts of regress problems: Russell’s Paradox, the Set of All Sets, etc."
How so? I agree it leads into all these things (Godel's incompleteness theorem too) and reflexivity is a key issue here, but *how* is infinity self-referential? please clarify!
 
 
cusm
22:28 / 08.04.03
I think an important if subtle point on the understanding of infinity is that it is not a term inherently of vastness, but of the unmeasurable. In this way, infinities can be understood as relative to the observer. That which is infinite in one frame of reference is not necessarily so in all frames. If I have a line that is longer than I can measure, it doesn't matter how long it actually is or how many other similar lines are added to it. Its still of unmeasurable length, and thus infinite to me. However, this definition does not exclude the possibility that by changing my frame of reference in some way (such as by extending my awareness to the plane of imaginary numbers, for example) that it could then be measured.

As for the concept of motion inherent in infinity, this is just one way something can be unmeasurable, and I think more an example than a rule.
 
 
—| x |—
05:47 / 09.04.03
You mean it wasn’t straight forward the first time?!?



Anywho (we’ll do, will due, will do)…

I have to say, off the top, that I’ll try to articulate this as best I can, but I admit that it is not entirely clear and precise in my head either. Here goes…

We have a finite set of words, many of which have sense on their own: dog, park, hat, slew, yellow, chance, Coca-Cola, for a scant few. These sorts of bits of language have sense that is more or less definite in the minds of most of the English speaking community; thus, we appear to have a more or less adequate grasp on the corresponding referent that these bits pick out. We have a set of finite words that have relations to seemingly discrete parts of reality. Now, it appears as if humans have a strong disposition to accept that discrete things are in themselves finite, and so, we believe/think/feel that these finite words we use are in some way connected to finite things in the world.

Now maybe this might be a biological fact, but it could also be a psychological fact—hardwired and/or software, and it might even be that it simply is the way of the world; however, and which+ever/other/none/all, it does seem as if we exist with, to quote Lurid, “…finitely biased brains.” We use finite words in our relating to what we—for whatever reason—experience as more or less definite and finite forms of differentiation: we are seemingly finite beings in a seemingly finite world.

Although I have an intuition that all things are infinite (where the math is more metaphor than an attempt to justify, validate or otherwise “prove”—all seemingly finite and discrete lines, plains, and solids are “composed” of the same infinite number of points), we typically experience differentiated and seemingly finite forms, but we do not appear to experience any infinite forms. Certainly we can conceive of infinity, even in respect to some of the finite words which refer to apparently finite things, take colours for example—red as pointed out by Lurid above—they can come in a conceptually infinite variety of shades, and perhaps any patch of red we experience is all the infinite shades of red in one, but we seem to, in any particular experience of red, interact with something that we interpret as finite and more or less definite. We do not generally interpret any experience as having an infinite content, and so, unlike most other words, infinity does not relate to a particular experience. So what does infinity relate to? It’s kind of like Grant pointed out, “You're left with a kind of sense-only, maybe.”

Infinity has to refer to something, I mean, we have a sense of infinity, we seem, after all, to be discussing something here, but it doesn’t seem as if we are discussing something that we experience in any direct way. So the referent of infinity has to be itself.

Now Quantum, you’ve stated that you don’t buy into the whole sense/reference thing, but think of it in terms that I’ve tried to sketch out: differentiated and discrete forms make up our interpretations of experiences and we have finite words and phrases that bear some relation with a perceived external reality; however, we don’t appear to have any direct experience of infinity insofar as we interpret any given experience. Thus, when we talk about infinity, we can’t be talking about something we’ve experienced, and yet, we are talking about something. So, unlike many words which seem to have a more or less definite relationship with our experience, ‘infinity’ has no such relationship.

I think it is important to take note that infinity is not a number, it is a set of relations—a bundle composed of an unending number of components all of which stand in relation to one and other. Now, it seems as if all of our experiences are relations to “sets” or bundles which are composed of components (again, we generally assume and interpret the differentiated forms we interact with as being composed of a finite number of components), and yet, we don’t seem to experience any particular bundle as being infinite. As I said before, I think all our words create an interpretation of events that “shield our minds” from direct contact with an infinite bundle of relations. Thus, steeped in our finite interpretation of experience, the immediate experience of an infinite number of relations must go interpreted as only itself; that is, infinity can only be itself—it is self-referencing, or, if you want, the sense of ‘infinity’ can only be its sense.

Now Quantum, you said, “Words are sense-only. (as are numbers- you can't point to a thing in the world and say 'that is 5').” And I entirely disagree, well OK, not entirely, but I do disagree. Words might be thought of as “sense-only,” but to me that’s buying into the whole sense/reference dichotomy. As I’ve sketched out above, our words bear some relation to our experiences: they are part of our interpretation of reality (whatever that may be). Whether we want to call that relation “reference” or not is unimportant, but we can’t deny that there are relations between our words and our experiences. Further, I do think we can point to a “thing” in the world that is five. We can point to five differentiated seemingly discrete and finite things and we experience the bundle, set, or group of relations that is five. It is exactly like pointing to a shovel and using the word ‘shovel’ in relation to a bundle, set, or group of components that make up the apparently discrete shovel. Numbers are no more or less “fictional” than shovels!

Anyway, I’ve spent much too much time on this (but it has been a great trip to the gymnasium)—I have to clean the dwelling. I know this is still probably muddled and muddy, but I am happy to attempt to address further inquiries in an attempt to be more clear!

eZ

Phew!
 
 
Quantum
12:49 / 09.04.03
Hmmm. Thank you for the detailed response, a few things...

"it appears as if humans have a strong disposition to accept that discrete things are in themselves finite, and so, we believe/think/feel that these finite words we use are in some way connected to finite things in the world."
As humans we define what is discrete. I see one thing (e.g. a plate of chips) you see fifty things (a plate and 49 chips) which is right? Perhaps a bad example, but I hope you see my point. Counting is a human action of grouping things together, identifying something as 'one thing' is a matter of discrimination.
A better example- take a thing, count it- one thing. Cut it in half. Count it again- it's either two halves of a thing or two things. Cut those things in half, you have four things, or four quarters of a thing, or two things each cut in half... etc. etc.
Counting is a way of putting things in sets (thus the set of sets problem etc.) which is a human activity.

"Infinity has to refer to something, I mean, we have a sense of infinity, we seem, after all, to be discussing something here, but it doesn’t seem as if we are discussing something that we experience in any direct way. So the referent of infinity has to be itself."
What? Why? Replace 'infinity' in that sentence with 'chance' (one of your examples). Do we experience chance in any direct way?

"So, unlike many words which seem to have a more or less definite relationship with our experience, ‘infinity’ has no such relationship."
I am happy to buy into the sense/reference idea for convenience, if it is clear that the referent of a word is our experience and not the thing itself (contrary to Frege's interpretation). But Infinity *does* correspond to our experience. We can imagine a line that goes on forever, or the infinite divisibility of space etc. or take the idea of time as cyclic- it goes on forever.
I understand we don't have as clear a relationship with it as, say, yellow, but compare it to Chance instead. Or any abstract concept.

"I think it is important to take note that infinity is not a number, it is a set of relations"
I think of it as an ascribable quality, an adjective- 'x is infinite' like 'x is yellow'.
"the immediate experience of an infinite number of relations must go interpreted as only itself; that is, infinity can only be itself—it is self-referencing, or, if you want, the sense of ‘infinity’ can only be its sense"
That's not what I understand self-referencing to be. Not sure what the last part means either- sorry!

"Words might be thought of as “sense-only,” ....
Numbers are no more or less “fictional” than shovels!"

Compare- "One" and "1". You're saying "One" can be sense only, but "1" is as real as a shovel. I disagree.

Is Infinity more or less real than Love, Courage or Beauty?


Cusm- I don't agree, infinite doesn't just mean 'unmeasurable', it means without end. Unmeasurable in principle and not just in practice.
 
 
cusm
15:41 / 09.04.03
Ok then, our experience (sense) of infinity is that which is unmeasurable, while the principle (referent) is that which is without end. However, That Which Is Without End is a concept we are unable to prove, and only assume to be true. We can not tell the difference between something that seems infinite and something which is, if anything can actually be infinite. The principle of infinity is yet an unproven assumption, which I think is where the bits about self-reference come into play, for objective infinity exists only as a concept or mathmatical conveniance. You can't have a referent to infinity because infinity can not be measured, so you have only the sense.

Also, on the lengthy exposition of brains, words, and reality, consider this handy import from the magick side of things that sums it up neatly: "The map is not the territory."
 
 
Quantum
10:10 / 11.04.03
Agreed- we only have the map!
But in my opinion, that's true of anything (shovels, numbers, fish) because we only have our perceptions of the world, the reality we believe in is inferred from them and is thus a construct of our brains/minds. I am a hardcore subjectivist/idealist.
To adopt your position on sense/reference for a moment, if infinity is sense only because it has no referent, what else is sense-only? Presumably superman, Pi, my cousin Sheila, any fictional thing or mathematical convenience is sense-only.
 
 
Lurid Archive
11:27 / 11.04.03
Quantum: I think that much of what you are saying becomes less controversial when you explain yourself. You don't think infinity has a referent because you don't think anything has a referent? Similarly, from before you said irrational numbers were a fiction, but the reason you have is that all numbers (and indeed everything else) is a fiction?

I'm somewhat interested in the idea of infinity as the unknowable. This kind of description seems to ignore a lack of finitude which to me would be the essence of infinity. (Unless one takes the finite as synonymous with the knowable.) So, for instance, language is said to be finite while Godel's theorem is not, when in fact the only infinity in Godel is the (potential) infinity of language.

As for "proving" the existence of infinity...*potential* infinities abound. So the number of meaningful sentences you can construct in English or, more realistically, the number of comprehensible(?) books it is possible to write is not finite in number. So, isn't it infinite? The step from potential to actual is a bit mysterious, however. But perhaps this is only because you cannot justify the infinite with reference only to the finite.

But then, I probably can't justify the existence of the colour red with reference only to the colour blue. So perhaps this lack of "proof" is simply the product of an unquestioned acceptance of one concept over another.
 
 
jackamo
19:11 / 12.04.03
PHILIP K DICK >
11-17-80
God manifested himself to me as the infinite void; but it was not the abyss, it was the vault of heaven, with blue sky of and wisps of white clouds. He was not some ferign God but the Gid of my fathers. He was loving and kind and he had personality. He said, "You suffer a little now in life; it is little compared with the great joys, the bliss that awaits you. Do you think I in my theodicy would allow you to suffer greatly in proportion to your reward?" He made me aware, then of the bliss that would come; it was infinite and sweet. He said, "I am the infinite. I will show you. Where I am, infinity is; where infinity is, there I am. Construct lines of reasoning by which to understand your experience in 1974. I will enter the field against their shifting nature. You think they are logical but they are not; they are infinitely cretive."
I thought a thought and then an infinite regress of theses and countertheses came into being. God said, "Here I am; here is infinity." I thought another explanation; again an infinite series of thoughts split off in dialectal antithetical interaction. God said, "Here is infinity; here I am." I though then, of an infinite number of explanations, in succession, that explained 2-3-74, each single one of them yielded up an infinite progression of flipflops, of thesis and antithesis, forever. Each time, God said, "Here is infnity. Here, then, I am." I tried for an infinite number of times times; each time an infinite regress was set off and each time God said, "Infinity, Hence I am here." Then he said, "Every thought leads to infinity, does it not?" Find one that doesn't." I tried forever. All led to an infinitude of regress, of the dialectic, of thesis, antithesis and new synthesis. Each time, God said, "Here is infinity; here am I. Try again." I tried forever. Always it ended with God saying, "Infinity and myself; I am here." I saw then, a Hebrew letter with many shafts, and all the shafts led to a common outlet; that outlet or conclusion was infinity. God said, "That is myself. I am infinity. Where infinity is, there am I; where I am, there is infinity.
 
 
—| x |—
08:17 / 13.04.03
When it’s good to have a little Dick in your ear…

Thank you for the detailed response.

It’s my pleasure, ya’ c?

As for your first point, “As humans we define what is discrete,” I couldn’t agree more. In case of point, note that I wrote in my first post, “…it appears that it is more a function of the human mind which discovers/invents separate and discrete objects…” The point being, with respect to the passage you quote, and your examples illustrate the point: finite words related to what we see as finite things in the world.

With regards to your second point, if I flip a coin, then am I not directly experiencing what the word ‘chance’ refers to? In rolling a die, I have a sense that there is a one in six chance that a given number will come up, and this sense of ‘chance’ bears a strong resemblance to what appears to be occurring in my interaction with the world. Which is to say, there does seem to be many situations which give a more or less direct experience of chance. Not so with infinity, it would seem. And this leads into the third portion of your response.

The examples you use, “We can imagine a line that goes on forever, or the infinite divisibility of space etc. or take the idea of time as cyclic- it goes on forever,” are all examples that correspond to, as you say, imagined things: we do not directly experience a line that goes on forever, nor the infinite divisibility of space. Your examples, as you say, are conceptual. And while I would agree that our imaginings are in some sense “things,” I still don’t think that we actually hold the whole of any infinity, expressed as a concept, in our head at a given moment. The concepts themselves are merely markers, indexes, shorthands, etc. but they don’t really appear to us as infinite. At any moment our experiences seem to be interactions with finite things: the finite concept of an infinite line, etc.

I think of it as an ascribable quality, an adjective- 'x is infinite' like 'x is yellow'.

I suppose this is reasonable, but I don’t know how far I’m willing to think that there is an actual difference between a thing and its properties; put differently, I tend to think that things are groups of relations, and I’m not sure if subject-predicate distinctions remain on such a view. I mean, the distinction obviously exists grammatically, but I’m not sure it reflects the actuality. Anyway, this is a little off topic, so I’ll set it aside for now…

To get back, the idea that I was trying to convey about the self-referencing, using the sense-reference distinction, was that the reference of ‘infinity’ was its sense. Whereas most words seem to have a referent corresponding to their sense, ‘infinity’ cannot have the same relation: it can only refer to itself.

As to the last quotation, there is two different points there—I apologize for running them together! The first point was that if you want to think of words as sense only, then that is your business; however, to affirm sense and deny reference seems to buy into the sense-reference distinction, IMO. So the second point was about the “fiction” of numbers. I am saying that ‘one’ and ‘1’ refer to the same thing, in the same way that ‘white’ and ‘blanc’ refer to the same thing; moreover, I am saying that the ‘1’ picks out something in the world that is no more or less fictional than a shovel. What is your disagreement?

Also, I don’t think it is a matter of infinity being more or less real than Love, Courage, or Beauty—I never saw that as the issue. The issue that I was getting at is that we can directly experience instances of Love, Courage, or Beauty, but we cannot and do not, typically, directly experience infinity.

Further, ‘Superman’ has a reference, it is the comic character who comes from Krypton, flies through the air, is the alter-ego of Clark Kent, etc. ‘Pi’ refers to the ratio between a circle’s circumference and its diameter (a well defined reference, really). It seems that words that have a fictional or mathematical sense are still able to have a reference.

I'm somewhat interested in the idea of infinity as the unknowable. This kind of description seems to ignore a lack of finitude which to me would be the essence of infinity. (Unless one takes the finite as synonymous with the knowable.) So, for instance, language is said to be finite while Godel's theorem is not, when in fact the only infinity in Godel is the (potential) infinity of language.

Lurid, can you say more about this? I’m not sure I get the gist of what you are trying to convey. In what way are you interested in the idea of infinity as the unknowable? And how does “infinity as the unknowable…ignore a lack of finitude which…would be [its] essence”? Please say more!

Finally, I’m not sure if the “concept of motion” is “inherent in infinity.” But I think I see what is meant by this: there is always one more; thus, motion is implied? However, perhaps this relates to what was said about infinity being relative to the observer? It seems to us that there is a sense of motion in infinity, but perhaps to the thing itself, or to something or someone in a different frame of reference, it is unmoving, i.e., complete or static—perhaps like eternity: self-same and unaffected by changes in the world.

If we look to religious conceptions of God, we find that there is often a presentation of the divine as infinite and eternal. Along with this, there is also ideas regarding God’s immanence in the world, and the transcendence of it. Also, we get further notions of God as being both static and dynamic. It seems to me that what we have are dichotomized pairs that create tensions amongst themselves, and perhaps a way to attempt to understand what might be meant by words such as ‘infinite’ and ‘eternal’ is to consider the union between those words and the other two pairs. Argh…I suppose this is a whole other kettle of fish though…

Take it eZ
 
 
Crimes_Of_Fashion
13:45 / 13.04.03
Apples and oranges.

Borges pointed out that infinity corrupts deeper and faster then stuffing your asshole with the bones of orphans.

Semantics follows cultural trends just as the value of information does.

Just wait until this wave pases, the next one's about the new japanese action figure\anime tie in and it's asthmatic casualties.
 
 
Quantum
15:41 / 14.04.03
Lurid- I don't think anything has a referent, true, but I believe that because I don't think the Fregean idea of Reference has any validity. I can understand the acceptance of the watered down version where the word has a referent but we can only know the sense, but then what's the point of 'referents'? Before we knew enough about Venus, the referent of 'the morning star' and 'the evening star' were thought different things.
IIRC Frege used the sense/reference distinction to solve the problem of the intentionality of language, and believed language was not only in our minds but accurately reflected objective reality and had an existence beyond our usage of it. The sense/reference distinction being used here is somewhat different, accepting that language is only a human construct but keeping the idea of the referent as a thing-in-the-world. How is a referent connected to a word in this view? I would say as a landscape is connected to a map, yes? i.e analogically not intrinsically.
This matches my view of language as sense only- I think words point to/indicate/make you think of things in the world, be they superman or a shovel. The things might not be there, the world might not be there, but language is self contained and consistent (like numbers) and thus subjective and not objective.

eZ- "if I flip a coin, then am I not directly experiencing what the word ‘chance’ refers to?"
No. 'Chance' refers to our theory of probability that predicts events. One event doesn't give it a chance :-) to work. It's a concept in our minds not a thing in the world. We project it onto the world, it's not intrinsic to it.
"all examples that correspond to, as you say, imagined things: we do not directly experience a line that goes on forever, nor the infinite divisibility of space"
Look up at the night sky. See the dark bits between the stars? How far can you see?
Stand between two mirrors. See the infinite reflections?
Calculate Pi/enumerate all integers How long will it take?

"second point was about the “fiction” of numbers. I am saying that ‘one’ and ‘1’ refer to the same thing, in the same way that ‘white’ and ‘blanc’ refer to the same thing; moreover, I am saying that the ‘1’ picks out something in the world that is no more or less fictional than a shovel. What is your disagreement?"
Thus you are saying that mathematics is a language, subject to the arguments we are making about the nature of languages.

jus to quote the film Pi "12:15- restate my assumptions. 1) Mathematics is the language of Nature")
 
 
cusm
18:01 / 14.04.03
Quantum, to read into your argument on referents, is what you are not accepting more the objectivity of reality? For granted, if all we have is our senses, we can't actually prove objectivity, only infer it. In this view, referents can only be accepted as a theoretical concept rather than accepted as a form of truth. Is that really a shovel? We may never know...
 
 
—| x |—
12:47 / 15.04.03
Borges pointed out that infinity corrupts deeper and faster then stuffing your asshole with the bones of orphans.

And Burroughs once remarked, “Just look around, buster! Just look around.” And what is it that we see if we actually are serious about looking around?—The Garden of Forking Paths! “Semantics follows cultural trends just as the value of information does”: it would have to—information is language, “…and language is virus,” as ole William S. told it. I’d bet that “…stuffing your asshole with the bones of orphans” is a damn fine way to manifest a whole lotta’ viral corruption.

But enough of the Side Show Stealin’…

1) “Look up at the night sky.”
OK.
“See the dark bits between the stars?”
Yes.
“How far can you see?”
I don’t know—looking at only a dark patch I have nothing by which to make sense of distance.

2) “Stand between two mirrors.”
OK.
“See the infinite reflections?”
No. I can only make out a limited number of reflections. There are many, but I can only see so far—the images blur into one and other—I did not experience an infinite number of reflections.

(And I really did these two things—go try it!)

3) “Calculate Pi/enumerate all integers How long will it take?”

How about four minutes?

But only if we did not appear to be limited and finite beings, and were able to perform an unlimited number of acts. If this was so, then in first minute we could write out the first integer, 0. After another thirty seconds, we’d write out the second integer, 1. After fifteen more seconds, we’d write out the third integer, -1. Seven and half seconds after that, we’d write out the fourth, 2. Three and one quarter second after that, we’d write the fifth entry, -2. And we’d continue to write out each integer in half the time we’d spent writing the one before it. After two minutes, we’d been done enumerating the integers. Then we could do the same thing for ‘Pi’. Starting with first digit, ‘3’, in the first minute, and continuing with ‘1’ thirty seconds later, ‘4’ fifteen seconds after that, and so on. Four minutes—in fact, the amount of time it takes to do this is arbitrary: we could do it in one minute, one second, or one nanosecond—but only if we too are, in some way, infinite.

But we’re not, at least, we don’t typically appear to be…so we can’t do it. Even if we had the remarkable power to consciously mark off and remember one integer for every second of a life, and let’s suppose we live, in this one life, to be 47 before—for whatever ghastly reason—we die, even then we’d only mark off around 1, 514, 764, 800 of the integers. So we haven’t really experienced anything close to infinity in this one remarkable life. We wouldn’t come close even in a finite series of similar remarkable past and future lives. I mean, no matter how many lives we use to add to the above number we still end up infinitely short of infinity!

Thus, we only are ever able to have a conceptual experience of infinity—like in the above we have a finite conception of infinity. Actually, there are two infinities in a finite bundle of 185 words. Now, is it reasonable to assume that from that, or any other, finite bundle we have actually experienced infinity? We certainly don’t have the whole of either infinities in my mind—otherwise we’d know the final digit of Pi! So in 1) and 2), the things we can actually do, we have no experience of infinity, and in 3) we can only experience those infinities if we are already in some way infinite. In other words, we do not live infinity through any of those methods.

Anyway, “'Chance' refers to our theory of probability that predicts events”: I agree! However, I do think that “one event” does give it, as you say, “…a chance :-) to work”—more or less. If the event is the entire process of flipping the coin—an apparently finite and discrete portion of spacetime—then during that time an apparently finite and distinct being, often calling itself “I,” experiences its “…theory of probability that predicts events.” Let’s be clear: whether the theory is for better or worse—whether it is accurate or not—is not our issue; the issue is that during that apparently finite and discrete portion of spacetime—during the apparent motion of moments to a seemingly distinct and finite being that views itself as exclusive from a seemingly distinct and finite coin—there is a seemingly distinct and finite pattern (the moments that compose the flipping of the coin) which represents, to the “I” of a particular memory, the fifty-fifty chance of the coin landing heads or tails. So in this respect, we, as an “I,” live chance.

In other words, the finite concept in our heads, ‘chance’, bears a more or less definite relationship to a finite pattern of experience—it doesn’t matter if we project it or its really there. What I am saying is that there is nothing in our pattern of experience that we can “project” ‘infinity’ onto or “discover” ‘infinity’ through—other than itself.

And yes, I am saying that mathematics is a language. “I am saying that ‘one’ and ‘1’ refer to [or bear a more or less definite relationship to] the same thing [or pattern of experience], in the same way that ‘white’ and ‘blanc’ refer to [or bear a more or less definite relationship to] the same thing [or pattern of experience]; moreover, I am saying that the ‘1’ picks out something in the world that is no more or less fictional than a shovel.” Again, what is your disagreement?

For granted, if all we have is our senses, we can't actually prove objectivity, only infer it.
~cusm.

Yeah, exactly, and I get the sense that Quantum agrees (but feel free to correct me, Q). But on the other hand, I can no more prove that “we” have senses: I can only infer from my own senses that the rest of “us” have senses; that is, as an “I,” we can neither prove a shared objectivity nor subjectivity, but can only infer these.

“Is that really [an eZ]? We may never know...”

 
 
Quantum
13:27 / 15.04.03
cusm- yes. I don't believe in an objective reality, I am a hardcore subjectivist/idealist.

"In other words, the finite concept in our heads, ‘chance’, bears a more or less definite relationship to a finite pattern of experience—it doesn’t matter if we project it or its really there. What I am saying is that there is nothing in our pattern of experience that we can “project” ‘infinity’ onto or “discover” ‘infinity’ through—other than itself." eZ
So what you're saying is we can't directly experience infinity? That's fair enough (religious experiences aside, that is indeed another kettle of fish) we can only experience too-many-to-count or too-small-to-see due to our finite senses. But that is enough for us to have a conception of the infinite. We can think about it even though we can't directly experience it (like the dark side of the moon) and invent a word to label that concept- 'infinity'. That doesn't make it self-referential.
Also I would say we can't directly experience a lot of things, what makes infinity special?

Mathematics- A shovel is tangible, an object, mathematics is a language, which reflects relations between things. If I believed in an objective world I would say the shovel is more real than mathematics, which is a human construct that only exists in our heads. A shovel is more real than the word 'shovel', thus a shovel is more real than the number '1'.

Since you think of things as bundles of relations I'm assuming you won't agree
 
  

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