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GOBY
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PostPosted: Wed Sep 01, 2004 7:37 pm    Post subject: Reply with quote

shurik wrote:
The problem is the I/O layer connecting brain to the physical universe. It's broken in so many ways it's not funny. If quantum mechanics is correct, the I/O can't even be fixed. On the other hand, the brain can internally recreate the universe of logic any time it feels like, which is why we can be much more certain about it.

...but..but.. <pisses pants> the "we" that you are talking about really only applies to the individual and his own apriori reality. As soon as there's a "we" interaction, there's a mandatory pass through the faulty interface. Of coarse I/O is both input and output, i.e. we screw up on both perception and expression, but that says nothing about which (the inside or outside reality) is more certain.
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tetromino
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PostPosted: Wed Sep 01, 2004 7:53 pm    Post subject: Reply with quote

piffle wrote:
Quote:
We apprehend the truths through logic (don't ask me how we can do logic, I don't know the first thing about how brains work). But even if we lacked the capacity to do math, it would still be there, and Alpha Centaurians or the Orion Machine People would still be able to derive the same theorems.

This relies on the assumption that the rules of logic are unambiguous and unassailable. But what constitutes acceptable logic is a matter of contention. Furthermore, attempts to found mathematics on logic alone all failed. Even Russell and Whitehead eventually conceded this point.

Are you saying that the axioms of set theory, the axiom of choice, Peano's arithmetic, the boolean NAND function, and the universal quantifier are not enough to derive pretty much all modern math? I am genuinely interested.
G0BY wrote:
As soon as there's a "we" interaction, there's a mandatory pass through the faulty interface.

I am not sure. If logic works the way I think it ought to work, any individual can at any time start with the basic axioms and rules of logic, and independently arrive at the same conclusion as anyone else, without actually communicating with them (other than to transmit the axioms). So as long as I am reasonably sure you received the axioms and your brain isn't buggy, I think I should be able to conclude that you are on the same page as me in terms of math.
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BaronVonOwn
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PostPosted: Wed Sep 01, 2004 7:54 pm    Post subject: Reply with quote

jetblack wrote:

Can you give me a convincing reason to argue with someone who does not accept the existence of truth?

Heh, that reminds me of a quote I culled from John Locke's "Enquiry Concerning Human Understanding":

"For I think no body can, in earnest, be so sceptical, as to be uncertain of the Existence of those things which he sees and feels. At least, he that can doubt so far, will never have any Controversies with me; since he can never be sure I say any thing contrary to his Opinion." -- John Locke
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papal_authority
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PostPosted: Wed Sep 01, 2004 8:07 pm    Post subject: Reply with quote

piffle wrote:
not everyone (the constructivists in particular) accepts material implication as a valid means of logical deduction

Fair enough, but that's not what I'm getting at. I'm saying that they agree on what material implication is and as such can check a proof using it. My wording was rather poor and I think it came across as more of a broad statement then what I intended :(

shurik wrote:
Personally, I would put it the other way around: mathematics, wave functions, logic actually exist in some objective fashion. Meanhile, that which people usually call "reality" is actually deeply uncertain, subjective, and unknowable.

Amen brother :D
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piffle
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PostPosted: Wed Sep 01, 2004 8:18 pm    Post subject: Reply with quote

Quote:
Are you saying that the axioms of set theory, the axiom of choice, Peano's arithmetic, the boolean NAND function, and the universal quantifier are not enough to derive pretty much all modern math? I am genuinely interested.


Is set theory really part of logic? In any event, the (already contentious) Axiom of Choice is not sufficient. To avoid Russell's Paradox (The Barber's Paradox, or Epimenides Paradox, i.e., the existence of impredictable sets) Whitehed and Russell were forced to include the "Axiom of Reducibility" and the "theory of types" to keep a strict hierarchy of propositions. This axiom recieves criticism because there is essentially no justification for it, other than it gives the desired result. The Axiom of Extensionality required to admit infinite sets also recieved similar criticisms.

To quote the historian Morris Klein:
Quote:

When Russell started his work in the early part of the century, he thought that the axioms of logic were truths. He abandoned this view in the 1937 edition of Principles of Mathematics. He was no longer convinced that the principles of logic are a priori truths and hence neither is mathematics, since it is derived from logic.

It should be noted though, that in order to even claim that mathematics is derived from logic, you really have to add in many things that really seem like they are outside the aegis of pure logic. In addition, people have also questioned how "non-arithmetical" parts of mathematics such as geometry and topology follow from logic.
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tetromino
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PostPosted: Wed Sep 01, 2004 10:07 pm    Post subject: Reply with quote

piffle wrote:
Quote:
Are you saying that the axioms of set theory, the axiom of choice, Peano's arithmetic, the boolean NAND function, and the universal quantifier are not enough to derive pretty much all modern math? I am genuinely interested.


Is set theory really part of logic? In any event, the (already contentious) Axiom of Choice is not sufficient. To avoid Russell's Paradox (The Barber's Paradox, or Epimenides Paradox, i.e., the existence of impredictable sets) Whitehed and Russell were forced to include the "Axiom of Reducibility" and the "theory of types" to keep a strict hierarchy of propositions.

I am unfortunately unfamiliar with Russel's work, so I can't say if he considered set theory to be a part of logic or not. His axioms of set theory are, as I was taught,
0. Primitive notions like "set", "collection", "function", "element of", "equal to", etc.

1. Extension: Two sets are equal iff they have the same elements
2. Specification: Given a predicate and a set, a new set can be constructed which contains elements of the other set that satisfy the predicate.
3. Pairing: Any two sets are contained in another set
4. Union: The union of any collection of sets is a set
5. Powers: each set has a corresponding "power set" (which is actually a collection, not a set)
6. Infinity: There is a set containing the empty set and the successor of each of its own elements (successor of S is defined as union(S, {S}) )
7. Substitution: there exists a function F with domain A and F(x) = {b : P(a,b) } given that P(a,b) is a predicate and { b : P(a,b) } exists for any a in A (I don't know how to phrase this more succintly).

Set theory phrased this way avoids paradoxes.

Suppose you take these axioms and add: the axiom of choice; NAND and the universal quantifier (which I think is enough to give first order logic); and Peano's arithmetic (which gives induction; actually, it might be derivable from set theory but I am not sure).
Again, I ask: are these things enough to derive math?

piffle wrote:
It should be noted though, that in order to even claim that mathematics is derived from logic, you really have to add in many things that really seem like they are outside the aegis of pure logic. In addition, people have also questioned how "non-arithmetical" parts of mathematics such as geometry and topology follow from logic.

Topology follows directly from set theory. Geometry can be developed in a number of ways; for instance, you can start with Euclidean space (derivable from set theory) and do geometry analytically, which doesn't require any extra axioms.
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piffle
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PostPosted: Wed Sep 01, 2004 10:58 pm    Post subject: Reply with quote

What reason is there to suppose the axiom of infinity, other than it gives the results that are desired?
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Suppose you take these axioms and add: the axiom of choice; NAND and the universal quantifier (which I think is enough to give first order logic); and Peano's arithmetic (which gives induction; actually, it might be derivable from set theory but I am not sure).
Again, I ask: are these things enough to derive math?

I don't believe this list is complete. In order to banish impredictive set definitions, Russell found it necessary to introduce theory of types, which ordered predicates into a hierarchy, so that they may only speak of objects and predicates at lower orders. However, in order to support induction, it was necessary to introduce the Axiom of Reducibility which allows for the collapsing of the predicate hierarchy. This was also criticised (by such as Weyl and Poincare) as arbitrary, and for having no justification beyond "it gives the result we want."
Quote:
Geometry can be developed in a number of ways; for instance, you can start with Euclidean space (derivable from set theory) and do geometry analytically, which doesn't require any extra axioms.

I'm not sure that Projective Geometry can be so developed (but I am not a geometer).


None of this touches on my point, though. Rather, even if all of mathematics were derivable from logic (appropriately burdened so as to be able to support such a claim), unless the rules of logic are a priori truths, neither is mathematics.
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