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 Post subject: Semantic clarification time!
PostPosted: Fri Nov 12, 2004 1:13 pm 
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Oh, great, and this is on a new page. Well, since it makes no (read: even less) sense without the previous post, go read that first. Go, go! For the love of the children!


One thing I realized when I was writing that post but was too sleepy/lazy to go into, is that there are multiple ways to define “complete understanding” of the universe, so it helps to know which one we’re talking about. (Note: when I say “we” from now on, except in (1), I’m referring to a hypothetical godlike descendant which would probably be the universe itself by that point) Here are all the ones I can think of:

1. A complete understanding of the workings of the mechanisms which underlie all physical processes. “The end of physics”. We would understand all of the laws of physics and would never be able to discover more, but we could not predict the future of the universe except in general terms. This seems like it might be possible even for a finite being, although I secretly hope for aesthetic reasons that it isn’t.

2. The most complete understanding of the physical universe that is possible, which turns out to fall short of (3). At this point we would know everything which can be known, due to inherent limitations of the universe we were unable to transcend.

3. Complete understanding of the physical universe. Knowing the position and momenta of every elementary particle in existence, or more realistically, the probabilistic description of such. The ability to predict the future or the probability of possible futures of the universe, including our own actions and thoughts. Requires an infinite amount of computational power, both to do the probability calculations and to get around the “Gödellian” self-consciousness problem. Probably equivalent to Tipler's Omega Point.

4. Complete understanding of the physical universe as well as all nonexistent things. Complete explicit knowledge, for instance, of the infinite set of all complex numbers. Complete knowledge of all the possible varieties, mating habits, etc. of Invisible Pink Unicorns™. Might require an amount of computational potential equivalent to Cantor’s Absolute Infinite, or something beyond that. Not only would we know every thought we would ever have, but have complete knowledge of every thought that could ever exist or be implied, including full understandings of hypothetical universes completely different from our own, including infinite, non-quantized ones. Ability to disprove (or prove) the existance of God, or more likely just become ver.

5. Something beyond (4). I have no clue, but I like to keep my options open :)


Of course, these change a bit depending on whether or not the universe is finite and quantized. But as far as I can tell it would just be changes from one type of seemingly-insurmountable-infinity to another, with the possible exceptions of (1) and (2).


Just wanted to sneak this in. Helpful explanatory notes in the previous post later, lunch now.


EDIT:
So, to clarify- at first it seemed like you were saying we could never gain type (1) understanding, with the Clarke quote about the weirdness of the universe. Then you used the idea of humans only being able to gain knowledge at a finite speed, to which I responded with hypothetical methods that would take us to (3). Then you brought up the complex numbers, which would require type (4) understanding to get around. (Although I then questioned whether type (4) is really a fair measure for "complete understanding of the universe", given its reliance on imaginary entities (see below))

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Last edited by Wandering Idiot on Sat Dec 04, 2004 4:29 pm, edited 2 times in total.

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PostPosted: Sat Nov 20, 2004 1:25 am 
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With respect the cardinalities of infinity:

The size of the set of natural numbers is denoted aleph-zero. It is quite easy to prove that any infinite set must have at least as many elements as the set of natural numbers. In other words, there is no infinite set smaller than the the set of natural numbers.

Any set for S which an onto function f: S -> N can be defined also has size aleph-zero. For example, the set of integers is the same size as the set of natural numbers, because we can define an f such that

f(0) = 0
f(x) = -2x, if x < 0
f(x) = 1 + 2x, if x > 0

We can even see that the set of rational numbers has size aleph-zero, and the rationals are not countable as are the integers and naturals. I define the set of rational numbers as the set of pairs (n,d) where n is any integer, d is any positive integer, and the greatest common divisor of n and d is unity. (That is, the fraction n/d is reduced.)

Consider only the subset of the rational numbers for which n is non-negative. (The negative rational numbers can be brought into the fold by a construction similar to the one I used above for the integers.) Define f in this triangular fashion:

Code:
f(0,1)=0
f(1,1)=1
f(1,2)=2  f(2,1)=3
f(1,3)=4  f(2,2)=*   f(3,1)=6
f(1,4)=7  f(2,3)=8   f(3,2)=9   f(4,1)=10
f(1,5)=11 f(2,4)=*   f(3,3)=*   f(4,2)=*   f(5,1)=15


In general, for n and d both positive,

f(n,d) is not defined if n/d is not a reduced fraction, but

f(n,d) = n + d(d-1)/2, otherwise

Obviously, the image of f (that subset of the natural numbers that can be generated by the function of f over all suitable pairs of n and d) does not contain all natural numbers, but it is an infinite set; and we know already than there is no infinite set smaller than the natural numbers, so the size of the set of non-negative rational numbers is the same size as the size of the set of natural numbers. Again, by the even-odd construction above, we can show that the size of the entire set of rational numbers is the same as the size of natural numbers.

But.

There are sets for which such an f cannot be constructed. One of those sets is the set of real numbers. There are lots of ways to represent the real numbers; for the purpose of showing that there are more of them than there are natural numbers, I will use the infinite decimal representation. That is, a real number has an integer whole part, a decimal point, and an infinitely-long string of digits representing its fractional part.

Now, there is the same number of numbers in the entire set of real numbers as there are numbers in the range 0.000... to 0.9999... so I will content myself with counting that latter set. (Why? Think arctan.) In other words, I am only really interested in the infinite-length sequence of digits after the decimal point. Presume for a moment that I actually could count the number of such fractions. Let me construct a possible table to represent such a relation:

Code:
0  0.010101...
1  0.12345...
2  0.31416...
3  0.21718...
.   ...
.
.


This relation is the most basic way to describe a function f relating the size of two infinite sets. However, there exists at least one fractional number q which is less than one and no more than zero that is not in that table. Let the whole part of q be zero, and let the (n+1)th digit of q after the decimal point be some digit that is different from the (n+1)th digit of the nth fraction in the table. For example, the first digit of the 0th fraction is 0; we choose 1 for the first digit of q. The second digit of the 1st fraction is 2; we choose 3 for the second digit of q. The third digit of the 2nd fraction is 1; we choose 9 for the third digit of q, and so on.

In this way, we can show that q is not equal to the nth fraction, for any n, because it differs from all of the n fractions in at least one place. Therefore the existence of the value q amongst the fractions shows that there are more such fractions than there are natural numbers.

(This proof, by the way, is called "Cantor's Diagonal". It is a very pretty proof, but it has a very big weakness: it doesn't produce an algorithm for generating anything. It is a non-constructive proof. In other words, it proves its statement, but it can't be used as a building block for discovering new facts.)

Again, but.

There are more real numbers than there are natural numbers, but we don't know how many more. It would be nice to assume that the real numbers are the next biggest set after the natural numbers, but we can't prove that fact. Nor, by contrast, can we find any set that is bigger than the natural numbers but smaller than the real numbers.

Worse.

We say that the size of the natural numbers is aleph-zero, and we allow that some set larger than the set of natural numbers has size aleph-one, whatever that is. (We do not know whether the real numbers have size aleph-one.) We cannot even say whether there are finitely or infinitely many different sizes of sets between aleph-zero and aleph-one. The proposition "there are infinitely many sizes of sets between aleph-zero and aleph-one" is known as the Continuum Hypothesis, and it has a very unusual and very disturbing property:

While it is a valid statement, it is nevertheless provably neither provable nor disprovable.

Anyway, all I wanted to say was -- as far as sizes of infinities go, we don't know how to count them beyond zero. We know that there are bigger infinities, we just don't know how to count them. Again in particular, the real numbers (and hence the complex numbers) do not necessarily have size aleph-one.


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PostPosted: Sat Nov 20, 2004 10:02 am 
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Tamayo, is the number of places after the decimal point definable?
For instance, can it be proved that the smallest possible number is, for instance 1*10<sup>-1*AlephZero</sup> (or some other infinite set)?

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PostPosted: Sat Nov 20, 2004 1:27 pm 
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In the infinite-string representation of the real numbers, the size of the infinite string is aleph-zero. If you can associate the elements of an infinite multiset (in this case, the digits of a real number) with the natural numbers, then that multiset is said to be "countable". In other words, all countable multisets have size aleph-zero.

The real numbers are said to be "smooth" in that the number of real numbers between any two real numbers a and b such that a is strictly less than b is equal to the number of real numbers. This is as a consequence of the fact that the real numbers are not countable. This same statement holds for the rational numbers, as well. (However, the reals are a continuous set, while the rationals are not.)


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PostPosted: Sat Nov 20, 2004 2:56 pm 
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Oww...my brain.

For some reason I've always had the vague notion (probably acquired from one of those engineering courses with statements qualified by, "Well, a mathematician wouldn't say this, but we're not mathematicians, so we'll just assume ...") that countability and set size were distinct attributes.

Is there a counter-argument for the position that the integers are countable but larger (twice)than the naturals? It seems you ought to be able to make the case that the counting index of your even-odd interlaced set goes to infinity twice as fast as the counting index for the naturals alone.

Of course, you still have a one-to-one relation ... Oww Oww Oww.

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PostPosted: Sat Nov 20, 2004 5:51 pm 
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In that a one-to-one relation exists, the sets are the same size, sorry. :-) It is meaningless to say that "counting indices go to infinity twice as fast" because all I am doing is showing the existence of a relation. I am not worrying about computational details. Mathematicians get to do that, unlike poor unhappy put-upon engineers.

Yes, the fact that if A and B are infinite subsets of the integers, and A is a proper subset of B, then A has the same size as B -- is weird. It's true, though, and the proof is fairly easy.


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 Post subject: And yes, I know I'm poking at the Platonism you mathematicians tend to hold so dear, but I'll live ;)
PostPosted: Tue Nov 23, 2004 6:06 pm 
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Heh, Tamayo proved me wrong about something (as far as the reals being definitely aleph1 - I've noted the correction in the relevant post). Silly me, I thought the Continuum Hypothesis was usually taken as tacitly true for the sake of convenience, but I was mistaken. Now if you'll excuse me for a moment...

*Commits Pedant Seppuku, messily*

*Admist the pile of entrails, a small nanite packet activates, replicates, and rebuilds the body*

Can't trust those alternate-reality versions until I make a post in this thread, what with people disparaging certain interpretations of quantum mechanics and all...


(Happy Fun Tangential Note: One thing I've been wondering that I can't seem to find a good answer to- maybe you could help me- what does an aleph2 infinity look like? If you assume the Generalized Continuum Hypothesis is true, I suppose it would be equivalent to 2^(aleph1), but that's hard to get an intuitive feel for*. What are its properties? What's a common example of one? Or is anything above aleph1 too vast for our feeble human intellects to make many definitive statements about? )


Now, getting back to the original line of discussion (voter turnout among the younger demographic? Fine, the previous line of discussion...) You never answered my point that in a completely finite and quantized universe, the full of set of reals doesn't actually exist in any meaningful way, and so it's no more valid to say that because it's not computable we can never understand the universe that it would be to say that we can never completely understand the universe because we'll never have full knowledge of the mating habits of Invisible Pink Unicorns™.

Which is to say, it's valid in sense (4) from my previous post, but that's a pretty far-out-there definition.


Now admit it! Submit to the possibility of us possibly reaching a point where there is, in some sense, no more possible knowledge of the universe to gain. (possibly) Is that so much to ask? :)


* Note for boarders who haven't goofed off at work today poking around articles on transfinite set theory: In set theory, 2^n (where n is a set of numbers) is equivalent to the set of all possible subsets of n, which as it turns out always has a higher cardinality than n itself. For infinite sets, this means it has a higher aleph number. (It just doesn't prove which aleph number it has, which is why I had to throw in the GCH above).

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PostPosted: Thu Nov 25, 2004 12:17 am 
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Ah, quick comment (I'll be back) --

The notation 2^A, where A is a set, is one way of denoting the powerset of A, that is, the set of all subsets of A. Another notation is \powerset{(A)}, which if you aren't using \LaTeX{} -- and you probably aren't -- looks sort of like P(A) but the P has to be in a script font.

By the way, exercise for the reader: prove that for any set A, the size of 2^A is larger than the size of A.


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 Post subject: I just realized my current sig is a quote by Bertrand Russell, which is eerily fitting for this line of discussion...
PostPosted: Thu Nov 25, 2004 1:30 am 
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Tamayo wrote:
The notation 2^A, where A is a set, is one way of denoting the powerset of A, that is, the set of all subsets of A. Another notation is \powerset{(A)}, which if you aren't using \LaTeX{} -- and you probably aren't -- looks sort of like P(A) but the P has to be in a script font.

Er, isn't that pretty much what I said? I mean, I left out the term "power set" since I know you already know it and it didn't seem like it was germane to the point for anyone else. (Then again, who am I to complain about unecessary tangential details ^^)

Quote:
By the way, exercise for the reader: prove that for any set A, the size of 2^A is larger than the size of A.

Cantor already did it :P

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PostPosted: Thu Nov 25, 2004 2:52 pm 
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Wandering Idiot wrote:
You never answered my point that in a completely finite and quantized universe, the full of set of reals doesn't actually exist in any meaningful way, and so it's no more valid to say that because it's not computable we can never understand the universe that it would be to say that we can never completely understand the universe because we'll never have full knowledge of the mating habits of Invisible Pink Unicorns™.


The full set of real numbers (indeed, complex numbers) must be necessary to describe the universe. Some time ago, there was a Big Bang. How long ago? Each reference point will indeed have a different answer to that question, due to the aforementioned warping of space-time, but the answers are nevertheless meaningful. They are also changing, continuously. The real numbers are the smallest continuous field.

Your hypothetical thing-that-understands-the-universe-completely must certainly understand at least how long it has been for it since the Big Bang occurred, or else it cannot make the same determination for any abstract object. It must then be able to compute the real numbers. QED.

The Invisible Pink Unicorns™ problem, while very important to all of us, is not important to the TtUtUC. The unicorns are not part of the universe. (Unless, of course, they have a gravitational, magnetic, strong or weak field with which to interact with the rest of the universe -- in which case, they're not as invisible as all that.) However, let's think of another thing that the TtUtUC can do that is beyond human power: answering the halting problem. The TtUtUC can predict exactly when Windows crashes next, even though no human can .... ;-)


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PostPosted: Sat Nov 27, 2004 10:25 pm 
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One day, I will convince someone to explain this thread to me in simple people terms.

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PostPosted: Sun Nov 28, 2004 1:16 am 
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Skjie wrote:
One day, I will convince someone to explain this thread to me in simple people terms.


Happy to! (Muahahahaha.)

Tamayo, misquoting Clarke or maybe someone else: The universe is not only stranger than we know, but stranger than we can understand.

Wandering Idiot: What if we had infinite time and space in which to understand the universe?

IcyMonkey: Knowledge is bunk. Truth is an illusion.

Tamayo: I deny IcyMonkey's premises and ignore what he says, making me a mean person.

Thinman: "The idea that the universe goes away whenever I shut my eyes is silly."

IcyMonkey: Things and representations of things are distinct and must not be confused. Effective representations are only that, and are no more true than any other representations.

Wark, Thinman: Yes they do. It's foolish to use a less effective representation.

IcyMonkey: Derrida is god. (Unfortunately, Derrida is dead now. Italics mine -- Tamayo) And again, blurring the distinction between the thing and the representation is an error.

Emy: We've wandered far off the initial subject.

Wark, Thinman: The truth of an analytic statement is a matter of symbolic manipulation, and thus mechanically determinable, and not dependent upon any representation of the world.

Madadric: How can you say that what you perceive is somehow not true, if at least for yourself?

Tamayo: Forget dictionary definitions of "true" and "false". Use these weird combinatorial definitions instead. (And there was much silence, and ignoring of weird combinatorial definitions of "true" and "false" -- Tamayo)

krylex: I know! Let's use dictionary definitions of "true" and "false"!

Herbal Enema: Here is a true statement: "Last time I went to Mars, I had a hot dog."

Wandering Idiot: (Taking us back to the original topic, "hows and whys" -- Tamayo) What if we have infinite time in which to discover things about the universe in which we live? In such cases, couldn't we learn all the infinite things there are to know about the universe?

IcyMonkey: You are all n00bs.

Emy: But we're old n00bs.

Ghost: I hate logic because logical statements describe only static states of affairs.

ptlis: Calm down, everyone.

Kakashi: It seems paradoxical to assert the truth of a statement like "truth is an illusion."

Ghost: No, what IcyMonkey is saying is that the distinction between objective reality, that is that which we think of as "truth", and subjective reality, that is that which we perceive for ourselves, is an error based upon the confusion between a thing and a representation of a thing.

Skjie: IcyMonkey, Wandering Idiot and Tamayo are blowhards. (Well ... yes. -- Tamayo)

Tamayo: (1) Don't accept arguments that contradict themselves. (2) In at least one case of which I am aware, the representation of a thing is actually a thing of the type represented. (3) Analytic statements are one instance of the class of statements that are objectively true, in IcyMonkey's sense. (4) Ockham's Razor is based on sound, necessary principles. It is more reasonable to accept the simpler of two sufficient explanations for a state of affairs. (5) There are infinite things to learn about the universe, but just because two sets are infinite does not mean that one set is not infinitely larger than the other. I think that of all the infinite sets, we human beings and indeed any intelligent creature I can even postulate are only really able to comprehend the smallest of those sets. One would have to be God to understand any more. Even were we immortal and able to live through any given time any number of times we should choose to do, we would still not learn everything there is to know about the universe.

Wandering Idiot: The real numbers have cardinality aleph-one.

Tamayo: No, they don't. We have no idea what the cardinality really is, only that it's bigger than zero.

Wandering Idiot: Anyway, consider the capabilities of a transhuman post-human intelligence, given immortality and time travel.

Thinman: Um ... are the integers bigger ... no ... my brain hurts.

Tamayo: Even given immortality and time travel, an being must have Godlike intelligence to have certain knowledge of any physical quantity.

Skjie: What in the hell is going on here?

Did this help? ;-)


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PostPosted: Sun Nov 28, 2004 12:36 pm 
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Which is more correct--Occam or Ockham? Or does nobody really give a shit?


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PostPosted: Sun Nov 28, 2004 4:43 pm 
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Ol' William came from a town in England named Ockham. However, he wrote in Latin, wherein the name of the town becomes Occamus. Either is correct, I would suppose. At least "Occam" saves a letter and is marginally easier to type. :-)


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PostPosted: Sun Nov 28, 2004 7:56 pm 
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The Baron wrote:
nobody really gives a shit


^_^

Seriously though, I'm pretty sure it's acceptable both ways.

And although it's hardly official, "Occam" does win out in the Googlefight, so more people probably tend to agree with Tamayo.


I like the summary, Tam. Now I'm tempted to go do the same for all the other philosophical debate threads (Although I'd probably have to do it in a more smart-assed way, i.e. "WI: But what about Teh Sooper-AIs?? OMG they're so smart! Tamayo: *tangential math ramblings* Icy: None of you really exist! Lalalalala!! Emy: *sex0rs Tam* ", etc.)

I'll try and come back later when I've got more time to see if I can explain myself better.

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PostPosted: Sun Nov 28, 2004 9:09 pm 
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WI wrote:
*sex0rs Tam*

Emy and Tamayo, in unison: Wait, what?! :o

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Emy wrote:
WI wrote:
*sex0rs Tam*

Emy and Tamayo, in unison: Wait, what?! :o

Everyone else: Yeah, just try to deny it.


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PostPosted: Mon Nov 29, 2004 12:51 pm 
The Baron wrote:
Emy wrote:
WI wrote:
*sex0rs Tam*

Emy and Tamayo, in unison: Wait, what?! :o

Everyone else: Yeah, just try to deny it.


::starts filming::


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PostPosted: Mon Nov 29, 2004 8:45 pm 
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Now that we're well and truly off-topic, how are IPUs both invisible AND pink?

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PostPosted: Mon Nov 29, 2004 11:24 pm 
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Wandering Idiot wrote:
Emy: *sex0rs Tam*

Um. It's not nice to call Emy stupid. She has much better taste than that. :-)

Thinman wrote:
Now that we're well and truly off-topic, how are IPUs both invisible AND pink?

They are so in similar fashion to the way colourless green ideas sleep furiously -- as an example of a syntactically-correct phrase that is semantically nonsense. That latter bit of nonsense is a fairly famous one, by the way, being Chomsky's own example.

ObOnTopicness:

Emy wrote:
Icy, the more i read what you're writing, the less sure i am that i ever understood what you seem to be talking about. Are you trying to rduce this to a semantic discussion, or a philosophical one?


Now to me, semantics are not trivial. Semantics are in fact essential to communication, and follow very well-defined rules. We are accustomed from grade-school education to syntactic rules for our languages, but definite semantic rules exist as well and are equally rigidly prescribed. The cliche of "quibbling over semantics" really disturbs me: for if what we mean by it is "arguing over the meaning of a sentence" then what it implies is a complete failure of communication, which is nothing trivial in the least.

However, IcyMonkey disagreed with me when he replied to Emy's question ...

IcyMonkey wrote:
A little bit of both, really. I don't think you can really seperate semantical discussion from "pure" philosophical discussion, as every single term or signifier we use comes loaded with all sorts of implications and connotations. Thus, redefining a word, or equating one word with another, can have some pretty big psychological/philosophical consequences. (I also don't believe it's possible to fully and completely seperate psychology and philosophy, though I agree that they can be treated as two seperate disciplines in many cases.)


If we are using a language whose semantics are well-defined and completely unambiguous, then we need no further discussion of it and can get on with discussing whatever else matters to us. Admittedly, English is not such a language. The predicate calculus, on the other hand, is one. In that fashion, then, psychology and philosophy are absolutely separable, and questions of *bleah* metaphysics just ... go away.

*washes mouth out with soap after having pronounced the M word*

Hmpf.


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