Then obviously you dont understand either Clayton, or Cohen, or both. Which is no surprise to anyone.
What mistake??
Of course. I’ve never once stated otherwise. Godel’s Incompleteness Theorems are not woe upon humanity… they are woe upon the project of The Perfect Language - a project that a large number of people still think has a chance of success.
But why are people reliable, I think is the unanswered question. The whole point of formalizing in the first place is because of the reliability problems of natural language. When it turns out that you cannot bootstrap a formal language from itself, then how does it help with the original goal of working around unreliable natural language by resorting back to (presumably unreliable) human common sense? I think that if you are the sort of person who is nervous about using natural language, you are also going to be nervous about relying on common sense.
Yes, because “proof” generally is not restricted to any particular formalism. But even Paul Cohen cannot prove that the axioms of set theory are consistent from those very same axioms. Godel proved that.
Clayton -
*shrug - OK
Based on scanning the article, I don’t think I have any issues with Frege’s ideas here except that I think there might be better ways to elucidate the problem nowadays than were available to him when he wrote (namely, it was not known that the brain is a neural-net computing device and so it makes sense to speak of information present or not present in the brain).
Universal by contrast to what? Do you know what lies beyond the Hubble sphere?
The claims in my sentence were:
-
Mathematics is a wholly human language
-
Mathematics is no better than any other domain of human language (I meant no closer to Absolute truth)
So, if you are claiming 1 to be false, please specify the aspect of mathematics that is not human.
If you are claiming 2 to be false, please specify how mathematics reveals any portion of Absolute truth, even one iota.
I have no idea what you’re even on about… how the hell could I possibly be denying anything?? I scanned the Frege article and I see nothing I disagree with. What next?
You got me, I’m exactly like a fat girl. Everything I’ve ever written hereby comes crashing to the ground.
crash
I was using the words as synonyms. In every other field besides philosophy, modern and contemporary are synonyms.
Clayton -
@Smiling Dave:
I see where the confusion is arising and it is my mistake. I wrote: “no matter how carefully you choose words and grammar - even if you choose as carefully as mathematicians choose them … the possibility of deriving a contradiction… is ever-present.” As stated, this is false. However, I meant this in the context of a regression argument, as Malachi has hinted at. If we formalize the system in which Cohen proved the consistency of the axioms of set theory (call it A), then we have a new formal system A’, whose consistency is unproven and, hence, uncertain. Ad nauseum.
Apologies for the confusion.
Clayton -
Is logic universal? Is it analytic?
Frege tries to demonstrate that arithmetic is analytic and on par with logic (that they are in essence the same thing). Logical objects (like thoughts) exist in nature and whatever the logical object is it can be known to us simply through our intellects. It is Universal unto itself due to logic (since logic is true in an analytic sense).
The point he makes aboutlanguage is that the Sinn is a logical object. All langauge emanates this. Formalization avoids ambiguity (this stuff is a given for further extrapolations).
Uh, you mean knowledge? “…is no better than any domain of human knowledge”?
The very nature of it. Sure, mathematics (the expression of it) is a human invention, but so is logic…what is being expressed is universal. Is mathematics going to be different past the Hubble zone?
Absolute truth as contrasted to what? Truth is truth. 2+2=4. Truth. I would dare say that that is a portion of truth. It was true 1000 years ago and will be true in another 1000 years. What about geometry? I envoke Frege due to his insistance that geometrical objects are graspable by our intellect and will not change from human perception, time, or place. In Sinn und Bedeutung he demonstrates this.
Which article did you go through? He makes a pretty good case for the formalization of language into logical form. Russell only expounded what Frege dealt with. Didn’t you vehemently deny this as a useful avenue of philosophy? The only reference to universal language that you are willing to entertain is Babel (which is a strawman). Your first point about elucidation is a demonstration of the problems of natural language that a logical language can do away with (particularly with geometry).
I knew it. You can shit and it won’t stink. Amazing.
If mathematics isn’t universal what would be the point in doing things like this and this? (to which you replied “WTF?”)
This is tiring. Universal by contrast to what??
Logic is a category of human knowledge - language, in particular. So, what is “universal” about it?
No, I mean what I said. Mathematics is a language. It originates in the intuitive theory of physics Steven Pinker discusses here. Specifically, mathematics is the intuitive theory sans the physics.
WTF??
Neither mathematics nor logic are human inventions. Both originate in the human brain which is of Nature’s devising. Even the specific conventions and notations of logic and mathematics are not of human invention (design), they are merely the byproducts of human action. This is the case for all human language (of which mathematics and logic are but one department).
As contrasted to relative, uncertain. Absolute truth is truth that is certainly true. Everything else is uncertain. But no human knowledge is certain. Hence, the intersection of human knowledge and Absolute truth is nought.
Frege was blind-sided by the Russell paradox. To construct the paradox, Russell used the very form of a diagonal argument which Cantor had earlier used to define powersets. Godel’s argument can be understood as an extension of this technique to provability (rather than truth).
It is well-established that human language cannot be formalized. Hilbert had set the vision of formalizing mathematical proof, a project which has succeeded in one sense (general-purpose computers, automated proof-checkers) but failed in its most ambitious sense… to render the mathematician obsolete. We have proved that the mathematician is ineradicable and that human language simply cannot be completely formalized (at least, not by humans). If you want to understand why this is, please read anything by Gregory Chaitin or watch any of his lectures.
What is Babel? When have I ever mentioned Babel? I haven’t the first clue what you’re talking about.
Because we can see that the laws of physics hold within the local environment where we shot the Pioneer probes into… if there is an intelligent alien species out there, they will have evolved subject to the same laws of physics that we are subject to. How does anything “universal” or “absolute” fall out from that??
Clayton -
To the OP, I’m just going to put these here:
Maynard, if you’re going to try to put this thread back on track, so will I:
http://www.youtube.com/user/flame0430/videos?view=0
Flame0430’s channel. All of the Bryan Magee interviews. ALL OF THEM.
Can you please explain this statement? I’ve never said anything even remotely like this so I don’t understand what prompted you to say this.
Clayton -
“the lanugage of babel”
Genesis 11:5, 6, 7, 8, & 9
Referring to the book that you cited earlier involving the quest of unviersal language. The lanugage of Babel looked to be the myth that they are referring to not that of a philosophical language.
Oh, you mean Umberto Eco’s book - Eco comes down squarely against the possibility of any universal language and that is, in fact, the whole point of his book. I haven’t read it (where am I supposed to find the time to read all these books!) but it comes highly recommended.
Clayton -
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