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petsfed
Apr 25, 2006, 9:19 AM
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I'm writing a paper on the subject, but it will be turned in by the time I look at this again, but its an interesting subject nonetheless. Do groups really exist, or do we group things as a mental abstraction because thinking about all the individuals at once is too difficult/impossible for our brains? Secondarily, with that in mind, do numbers other than one and zero actually exist, or are these once more abstractions to make the thinking easier?
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tradman
Apr 25, 2006, 9:21 AM
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Wow! Heavy duty philosophy!
I'm sure largo will be able to give you a really great answer to this!
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unabonger
Apr 25, 2006, 11:41 AM
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So if something is an abstraction it might not exist?
What are the qualifications for "existence"?
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jackflash
Apr 25, 2006, 11:52 AM
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I had a whole class on composition last year. Fun stuff. We primarily focused on Trenton Merricks Objects and Persons in which he concluded that objects do not exist. What exist are objectlike arrangement of parts. In other words, there are no baseballs, there are only atoms arranged baseballwise. He had an interesting argument for the position too, which I'll probably slaughter from poor memory.
Grossly simplified, a baseball involved in some event, say, breaking a window, is causally irrelevant to its parts acting in concert breaking the window. The parts acting in concert break the window. The event is not determined by more than one cause. Therefore the baseball did not break the window. Thus, if the baseball exists, it has no causal power, and the baseball is not an object because objects have causal powers.
Believe it?
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petsfed
Apr 25, 2006, 2:57 PM
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In reply to: So if something is an abstraction it might not exist? What are the qualifications for "existence"?
No, an abstraction is a mental construct that does not exist outside of our minds. We should not be convinced something exists if its existence would in no way actually affect the state of affairs, nor would its absence.
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brent_e
Apr 25, 2006, 3:11 PM
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In reply to: I'm writing a paper on the subject, but it will be turned in by the time I look at this again, but its an interesting subject nonetheless. Do groups really exist, or do we group things as a mental abstraction because thinking about all the individuals at once is too difficult/impossible for our brains? Secondarily, with that in mind, do numbers other than one and zero actually exist, or are these once more abstractions to make the thinking easier?
your first quetion is pretty good.
Your second question is "useless," I think. how did you answer that and what direction did you take to answer that??
take care
Brent
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yanqui
Apr 25, 2006, 3:11 PM
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In reply to: I'm writing a paper on the subject, but it will be turned in by the time I look at this again, but its an interesting subject nonetheless. Do groups really exist, or do we group things as a mental abstraction because thinking about all the individuals at once is too difficult/impossible for our brains? Secondarily, with that in mind, do numbers other than one and zero actually exist, or are these once more abstractions to make the thinking easier?
Not all philosophy is ontology, but ontology is kind of trippy. Mathematicians play the ontology game all the time, proving stuff does or does not exist. It's part of the deal.
I hate to play 'name that philosopher' but someone who knows the history of philosophy is gonna recognize where I'm starting from, so I might as well drop names. Anyways, my own opinion is that Kant was on the right track when he tried to argue that, to some extent, what you seem to be calling 'groups', or better yet, what Plato called 'the forms' (i.e the forms of thought) are a necessary part of human knowledge. This is not exactly ontological in the old school sense. For example, Kant argued in the transcendental deduction that in order for there to be knowledge, human consciousness had to be able to distinguish dreams from reality, which in turn depended on the ability to apply concepts like 'space' and 'time'. Without concepts such as these, there could be no human knowledge. What I like about this is that it is not an attempt to prove that abstractions like 'space' and 'time' exist apart from from our consciousness of them, but rather to show that concepts such as these are a necessary part of human knowledge of the world.
At any rate, one problem with Kant's conception, in my opinion, is that he had a very rigid idea about the forms themselves. And this coming from a mathematician (me!). Undoubtably Kant's conception is not unrelated to the Newtonian-physics-style world view that permeated the academic culture of his time. To Kant, the concepts like 'space' and 'time' seemed to be sort of hard-wired into human consciousness, when in fact, I would say concepts like these are somewhat fluid and a result of a shared, cultural evolution. How rigid is our concept of space? Just 200 years ago a great mathematician like Gauss denied the possibility of non-Euclidean geometry, Now non-Euclidean geometry is a basic part of the generally (acdemically speaking) accepted global world view (i.e. Einstein's general relativity). One possible unified theory entails that the universe has 11 dimensions.
Even mathematics, probably the most rigid of all our conceptual constructions, evolves and changes. For more than 2000 years, since Euclid, one basic idea of the nature of math was that it was a closed system that proceeded from a few 'evidently true' and generally accepted premises to a complete theory. This closed system was often held up as model for all kinds of knowledge (e.g. philosophical). Just get the right list of premises and then everything would follow with formal, deductive precision. We now know this conception was, to some degree, wrong. In fact, it's been shown that any formal theory sufficiently powerful to include the real numbers, necessarily entails an infinte list of true and unprovable statements (i.e. premises, in the old view). How do we choose among competing premises then? Outside of the criteria of consistency, this is a metamathematical question. The reality of the situation is that these aspects, as well as the concepts mathematicians consider important (e.g. 'topology' or 'ring theory' or 'Riemannian geometry') are aspects which have gradually evolved in a shared culture of mathematicians. For sure these 'forms' are a necessary part of our mathematical knowledge. But I do not believe they are somehow imposed on our consciousness in some inevitable way. On the other hand, we are not totally free to choose these forms isolated from the context of our cultural evolution.
A related problem is the following. A sort of thought experiment, if you will, ala Einstein's thought experiments about riding on a beam of light. Suppose we were to meet intelligent beings who in a physical sense were quite different from ourselves, e.g. intelligent insects. What I mean by 'intelligent' is that these insects had some sort of shared, evolved culture which allowed them to interact with the world in a way we could recognize as 'intelligent'. Now suppose they had developed a science, a mathematics and a technology sufficiently powerful for say, interplanetary space travel. It's not hard for me to believe that their mathematics would have some sort of structure obviously isomorphic to the integers. Counting and measuring are a rigid basis of mathematics for obvious reasons. In this sense, I would say that the concept of number (in the sense of counting numbers), like you propose above, is a very rigid part of science, mathematics and technology.
But now you might ask: would these insects have developed algebraic topology? Would they localize communative rings? Would they know about finite projective geometries? Would they study Lie group representations? These are topics that any Ph.D in mathematics is aware of, but it's certainly possible to imagine that the more advanced aspects of our intelligent insects' mathematics would be so different from ours as to be virtually incomprehensible even to a Ph.D. (and vice-verse).
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cellardoor
Apr 25, 2006, 4:10 PM
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yanqui,
Nice post! but quick question. You said:
In reply to: In fact, it's been shown that any formal theory sufficiently powerful to include the real numbers, necessarily entails an infinte list of true and unprovable statements (i.e. premises, in the old view).
Do you have a paper on this that you could link to or perhaps sketch for me a quick argument for this idea? I'm not doubting your assertion in the least i merely would like to seriously know how this is shown as i've always viewed math as the closed system you mentioned.
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yanqui
Apr 25, 2006, 4:37 PM
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In reply to: yanqui, Nice post! but quick question. You said: In reply to: In fact, it's been shown that any formal theory sufficiently powerful to include the real numbers, necessarily entails an infinte list of true and unprovable statements (i.e. premises, in the old view). Do you have a paper on this that you could link to or perhaps sketch for me a quick argument for this idea? I'm not doubting your assertion in the least i merely would like to seriously know how this is shown as i've always viewed math as the closed system you mentioned.
Sorry I don't have more time, but class calls. Here's a Wikepedia reference. Lots more are available. I wanna mention an interesting point about Gödel's incompleteness theorem: Roger Penrose tried to use this theorem (perhaps succesfully) to prove that the human mind is not a digital computer. Mind blowing stuff.
http://en.wikipedia.org/...completeness_theorem
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petsfed
Apr 25, 2006, 6:25 PM
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Just finished the damn paper. For what its worth, its based off of three essays:
"On What There Is" - W.V. Quine
"Empiricism, Semantics, and Ontology" - Rudolf Carnap
"What Numbers Could Not Be" - Paul Benacerraf
Basically Benacerraf claims that numbers can not be sets, and we were supposed to determine if, in light of Carnap's rejection of "external questions" as non-sensical (and Quine's support of that position) if we can commit to the existence of numbers. I concluded that we could not, and the answers to the two questions composed my thesis.
Sorry, what I meant with "groups" are like 3 wrenches is a group of wrenches. Is the group a mental construct to make it easier to think about all three at once, or something intrinsic?
As for the second question, its just a corollary of the first. If it is the case that we use a mental construct to deal with more than one object at a time, can we make the case that the only numbers that are actual are zero (rather the absence of) and one (rather the presence of)? As soon as I get more sleep, maybe I'll explain my position.
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organic
Apr 25, 2006, 7:06 PM
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Groupings are natural. If you study neural networks and brain imitation most are based off of the fact that our brains generalize things. Object recognition makes this very clear. We call things car, plane, can, food, when in fact each object could be completely different in structure, color, &c than another car, plane, can food. I think groupings increase our perception to have macroviews of things and essentially groupings increase our thought speed and processing. A venn diagram can be a good way to think of this, overlapping groups can eventually bring precision.
I think you are focusing a little on semantics though. What makes something a group? What makes something individual? Are we not still looking for the great atom or indivisible? Everything is composed of other things and related to other things. I think the question you should be asking is does 1 or 0 really exist. If you think about it 1 is actually a group it is a point of reference that is related to 0. Computer programming even starts at the reference point 0 so 1 is actually the second reference point.
Of course if you want to argue semantics everything is just an arbitrary reference to create meaning.
Can you ever know what is 0? What does 0 mean? Unless you can define everything anything just has arbitrary meaning that is why it is so hard to make artificial intelligence. You can tell a computer what a cat is but how does it know a cat is not a dog is not a car, it must know what everything is. There is no exactness. That is why science usually uses approximates and statistics. Cause we don't know for a TRUTH. We use confidence intervals cause WE ARE CONFIDENT there is correlation.
To ask if something exists you must first have a definition of what it is? So first you must define 'group', '1' and '0'.
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arrettinator
Apr 25, 2006, 7:58 PM
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I'll get in on this one once it gets to be as big as the About God thread. :roll:
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petsfed
Apr 25, 2006, 9:03 PM
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In reply to: I think you are focusing a little on semantics though.
Well, that's the thing. Read the title of the Carnap paper. Modern philosophy is semantics first. Imprecision of terms leads to bad philosophy.
FWIW, my paper basically said this:
1) Numbers are simply a convenient way of saying something much more complex. We say three Xs, when what we mean is (X + X + X). Needless to say, that's a very ungainly construction, and very difficult to work with.
2) 1 and 0 are special numbers because they have additional meaning. 1 means the presence of something, 0 means the absence. In fact, one X means (X), while zero X means ( ).
3) We cannot reconcile the question "can numbers be sets?" with the following idea: The only questions that make sense are those that relate to the contents of our linguistic framework. Those that relate to the nature of the framework itself (so-called "external questions") are meaningless.
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organic
Apr 25, 2006, 11:29 PM
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In reply to: In reply to: I think you are focusing a little on semantics though. Well, that's the thing. Read the title of the Carnap paper. Modern philosophy is semantics first. Imprecision of terms leads to bad philosophy. FWIW, my paper basically said this: 1) Numbers are simply a convenient way of saying something much more complex. We say three Xs, when what we mean is (X + X + X). Needless to say, that's a very ungainly construction, and very difficult to work with. 2) 1 and 0 are special numbers because they have additional meaning. 1 means the presence of something, 0 means the absence. In fact, one X means (X), while zero X means ( ). 3) We cannot reconcile the question "can numbers be sets?" with the following idea: The only questions that make sense are those that relate to the contents of our linguistic framework. Those that relate to the nature of the framework itself (so-called "external questions") are meaningless.
Doesn't '22' mean something just as much as '1'? They both indicate the presence of something.
Would we be more lost with no '1' or no 'group'? Which is of more value?
In mathematical terms X*0 = 0; X^0 = 1. If '0' can modify is it actually an absence? It seems to have a value to me.
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unabonger
Apr 26, 2006, 12:25 AM
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In reply to: It seems to have a value to me.
Sure. But consider the value of 1,000,000 compared to say, 10. Those zeros represent a lot of value. The importance of zero is as a placeholder.
Hindu's had the jump on zero and were thus able to describe many natural phenomoenon, including the sun-centered orbit of the earth, 1000 years before copernicus and his marketing team.
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organic
Apr 26, 2006, 12:47 AM
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In reply to: In reply to: It seems to have a value to me. Sure. But consider the value of 1,000,000 compared to say, 10. Those zeros represent a lot of value. The importance of zero is as a placeholder. Hindu's had the jump on zero and were thus able to describe many natural phenomoenon, including the sun-centered orbit of the earth, 1000 years before copernicus and his marketing team.
My point exactly I totally agree with you! Zero doesn't seem to be an 'absence' as petsfed says.
Also another thing I thought of. Think of programming when you compare something to 0. Zero has a definite value. If something can be compared and has a 'value' mathematical or even philosophically can it be considered an 'absence'? Most consider 'NULL' an absence so to speak, but it is even not that.
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petsfed
Apr 26, 2006, 3:32 AM
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It took me a long time to muddle through a lot of what you're pointing out. I'm tempted to just post the whole damn paper, but the fact that I finished it after 25 hours without sleep kind of shows.
Anyway, when we say 1,000,000 dollars, are we considering a million individual dollar bills? Is such a thing possible? My claim is NO. I claim that any number attached to an object, that is numbers in common parlance, as opposed to mathematical parlance, is in fact a cognitive short hand so that we can actually think about it.
Returning to the wrench example, when we think about three wrenches are we thinking about each wrench individually and simultaneously, or as a group? Are we just reducing back to one thing, in this case a group of wrenches? That's what I indicated with the (X + X + X) bit. Three Xs is mental shorthand for that very phenomena.
The thing that's really crucial about all of this is that you HAVE TO understand the argument in the Benecerraf paper. He states, basically, that the number of members in a set has absolutely no relation to the rest of the set. So the question was, how can I put together an understandable ontology that doesn't rely on numbers to describe the size of the set, while still allowing for our numerical nomenclature to remain. The creative reorganizing, subdividing, as well as duplicating the set allows for a complete rederivation of arithmetic within our practical parlance.
What I eventually concluded is that numbers are just names for phenomena. We could just as easily call (X + X) cake as two Xs, its simply for convenience that we use the latter.
And returning to the 1/0 thing. Initially it seemed like a good solution to the problem, but it eventually became apparent that it wasn't, and I subsequently abandoned it. It did, however, lead me down a road that allowed me to suggest that absence and presence where the fundamental "numbers" (for what of a better word) of our practical parlance. They were easily represented as 0 and 1, respectively, but it confused the issue so one of the first things I did was to remove the numerals from further discussion.
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thegreytradster
Apr 26, 2006, 4:06 AM
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In reply to: I think I felt that before It was kind of an experience One shining moment lost in time The one with the showroom shine Well, I'm sorry girl Yes, I'm sorry boy But I feel that way all the time Till I finally made it Life was kinda hit or miss But after I made it Life was "Take a hit of this" And I'd love to talk philosophy
But I've gotta take a piss
Man, that philosophy runs right through ya
Has anyone here actualy made it all the way thru Roger Pendrose's "Road to Reality"?
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gritstoner
Apr 26, 2006, 8:55 AM
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i knew having a relation that was a latin teacher would come in useful sometime, especially when philosophy is concerned.
ego escendo, ego sum
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yanqui
Apr 27, 2006, 3:36 PM
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In reply to: Has anyone here actualy made it all the way thru Roger Pendrose's "Road to Reality"?
Why is it good? My familiarity with Penrose is the 'Penrose Twistor Construction' which had some applications in my particular area of mathematical research. There were a few papers published by bigwigs and I heard some talks at conferences, but I never studied it myself in any detail.
In fact, being a somewhat lazy-ass sum-bitch, I've never actually read anything by Penrose.
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