Posted on 10/25/2002 12:14:19 AM PDT by jennyp
Numbers aren't exactly my field of expertise either. While one could posit a universe where countability wasn't important for any practical purposes, it seems fairly obvious to me that meaningful observation requires a space that has practical countability. Cognition (in the abstract, not just human) can't develop in a space without all sorts of countable properties.
IMHO, first order logic is quite handy for computer science but fails in natural science and thus would not be of a necessity portable to another universe or domain – which would be subject to other physical laws.
For lurkers: there are various schools of mathematics and tortoise and I are at odds because I fall in the Platonist school.
Formalism is the view that mathematical statements are not about anything, but are rather to be regarded as meaningless marks. The formalists are interested in the rules that govern how these marks are manipulated. Mathematics, in other words is the manipulation of symbols.
Unfortunately, this philosophy was proven unfit by Gödel's incompleteness theorem [see below] ...
A break off of Formalists school is the Logicists school, championed by Bertrand Russell and Gottlob Frege, who sought to show that are knowledge of mathematical truth was as certain as our knowledge of logical truth. They attempted to define mathematics in the language of logic. Their efforts resulted in some important ideas, such as the relationship between number theory and set theory, but ultimately this enterprise was found faulty as well due to paradoxes such as Russell's Paradox, an important principle of set theory which could not be based on logic.
Inventionism
Inventionism, also sometimes called Constructivism, holds that true mathematical statements are true because we say they are. Mathematicians do not discover mathematics, as the Platonists claim, they invent new mathematics.
Intuitionism
The easiest way to define intuitionism is that it is the corollary of logicism. The Logicists want to define mathematics in the language of logic. The intuitionists want to define logic in the language of mathematics.
Platonism
The view as pointed out earlier is this: Mathematics exists. It transcends the human creative process, and is out there to be discovered. Pi as the ratio of the circumference of a circle to its diameter is just as true and real here on Earth as it is on the other side of the galaxy.
The above definitions are from What is Mathematics?
Incompleteness Theorum (Gödel)
But it did happen. So external proof is not required for an event to occur. Therefore, a person does not need external proof in order to believe an event occurred.
Well, of course. But believing an event occured, no matter how ardently, isn't the same as proving it. Most of what humans do, including in mathematics, proceeds happily without proof.
You wrote:
IMHO, first order logic is quite handy for computer science but fails in natural science and thus would not be of a necessity portable to another universe or domain – which would be subject to other physical laws.
I'd call this a bit of an over-reach. Predicate logic doesn't really "fail" in natural science. It just isn't called into the game very often. Occasionally, one sees a truth table imployed where we've made a system with so many conditionals it's hard to think about without turning our brains to putty, but for the most part, most daily reasoning, including most technical and scientific reasoning, takes place without the need of formal logic, because what most people think about most of the time isn't complex enough to require formal tools to avoid errors of logical conflict. We still obey the rules of logic (which apply pretty well to large, gross objects in our local environment) as laid down by Aristotle and Boole, we just don't deign to notice. In our own universe, we have examples of useful systems of logical thought (notibly in the sub-nuclear realm of quantum mechanics) which obey a different set of fundamental rules.
...
You wrote:
Unfortunately, this philosophy was proven unfit by Gödel's incompleteness theorem
I had to think about this for a bit--it seems to me that the formalist school hasn't suffered any worse than the rest of the big league logical sports teams. What Godel demonstrated certainly takes the wind out of Grand Project of formalizing all of mathematics. But this limitation is about equally sobering, in my opinion, for any players in the game. If I were betting on this event, I'd put my money against Platonists in this regard--in that we have now I would suppose, less of a clear vision as to what sort of ghostly reality math and/or logic represents. Just a hunch, of course.
At any rate "unfit" seems a bit strong. Hilbert's agenda of producting logical systems divorced from any domain of discourse (clearly a formalist agenda) proved, in the end to be highly fruitful, and is far from running out of steam as we speak. Establishing isomorphisms between disparate domains of discourse under any given set of logical rules has been a discipline that started as way to escape the type conflict dilemma, and the undecidability dilemma (which it failed to do) but in the end, has turned out to be quite helpful in sub-nuke and topology studies, amongst others.
Me too, PH. I'm so happy to find we agree on this! ;^)
Come on now, BB. Of course I agree that man is unique! We may be related to all the other life on earth, but we're certainly way superior. All that we disagree on is a few little intermediate steps along the way. That's not really so terrible, is it?
That's putting it rather starkly, A-G! But IMHO it's the truth of the matter.
Of course, there's nothing that says man has to rebel against God and the universe. Unlike animals, he has free will: He always has the option to choose the good.
The Greeks had a word for the good: Agathon. It's a symbol that encompasses the ideas of truth, justice, and beauty. You might say it represents the divine standard and measure of universal reality. By this standard and measure, man's "rebellion" against God turns out to be equally a flight from the ordered universe as it is.
Of course, it really doesn't matter whether a man actively chooses to rebel (that is, to do evil), or merely backslides into it. The result is the same in either case, IMHO: disorder -- of the personality, of family and social connections; of the natural world.
But maybe this sounds 'way too "simplistic."
Oh, just a "few little intermediate steps," eh? LOL!!! Okay if you say so, Patrick dear. :^)
I'm not so sure that it is a difference in schools of thought as it is a difference in mathematical backgrounds. I tend to look at everything in mathematics through the eye of Kolmogorov information theory and related fields of computational theory. When people posit things, I immediately frame everything in the context of those fields (which fortunately have very broad application and fairly penetrating theoretical value).
I'm pretty pragmatic about mathematics, probably because my real background is engineering and science, though I'm far better known for my applied mathematics work. My "school" lives somewhere between Formalism and Platonism. Incidentally, I don't really see how Godel's IT is a serious problem for Formalism, at least no more of a problem than it is for anyone else. There are many important theorems in other areas of mathematics that are analogous to GIT (including some extremely useful variants with respect to computation theory found in information theory). The work of Chaitin, Fisher, and others really puts a nastier limitation on our knowledge than Godel does in my opinion. Godel merely asserted that there was a limit, but others have shown exactly what the nature of those limits are and to the extent that we are regularly bumping up against those limits. Bertrand Russell's "Principia Mathematica" has been known to be a fool's errand for some time, at least in its original intent, and I don't think many people are working on a mathematical Theory Of Everything.
In the event you are trying to reduce cognitive experience to formal constructs – since we are blessed on this forum to have an expert in Artificial Intelligence - I am pinging him for his comments.
Heh. Ironically, I am quite probably the most qualified expert on AI theory on this board, though I don't spend too much time on it here. None of the rest of the guys that are recognized experts in the field are Freepers that I am aware of, and I am at least acquainted with most of them. And if any of them read my posts on AI, they'd be able to name me pretty quickly from familiarity with my theoretical work. :-) Interestingly enough, I know for a fact that there are a number of famous scientists and physicists who have been Freepers for a long time. A lot of really well-known and interesting individuals from the academic community hang out here incognito, including individuals we even occasionally talk about in threads -- heaven forbid it gets out that they are regulars on FreeRepublic!
Back to the topic, you can reduce "intelligence" in all meaningful forms to the same formal constructs (and some related proofs have been published in the last couple years regarding this), but not in the sense that most people imagine when they make the assertion that "you can't reduce cognitive experience to formal constructs". A lot of the really cool work is recent, and to a great extent, unpublished. The formal constructs that are emerging are extremely elegant, but not something you can explain to people in an elevator pitch. Explaining it to people who are very competent theoretically still takes several hours for me; people take considerable time to wrap their heads around the math despite its relative simplicity. They just aren't used to thinking about some things in the directions it takes you. Its good stuff, though, and just starting to produce really interesting results in practical application.
Thank you for the discussion, tortoise!
No, thank you! :-)
Orthogonal is better than the term Gould chose to distinguish between reason (and its realm of influence, science) and the spiritual dimension --- "non-overlapping magisteria." At least orthogonal shapes intersect.
In Gould's proposal a conflict arises. Invariably for him science occupied the deep end of Truth and religion was relegated to the separate, "non-overlapping" shallow end of Meaning.
Meaning must intersect Truth, else it wanders meaninglessly in the void.
These are challenging times for thinking, rational people who nonetheless recognize the limitations of reason. Theories of remarkable explanatory power are revealing us to ourselves in unprecedented ways. It sometimes seems as though we've got the puzzle of life knocked.
Insane criminals waging war in the name of faith make matters more difficult for people of all faiths. Sometimes --- oftentimes really --- it seems as though we are at the end of the Religious Age.
But for me the mystery of existence has proven profound enough to keep my pride in check. I take counsel from Santayana's description of Hegel: "He described what he knew best or had heard most, and felt he had described the universe." The same tendency surfaces often among many of our learned public intellectuals, especially in the sciences. I hope to avoid their condition. On rising each day --- this is strange, and probably foolish to admit, but quite true I assure you --- I ask myself as I stand before the mirror: "How did we get here? How did all this come to be?" After many years of starting my day with those questions, and much investigation, I can safely say that I'm no closer to the answers than I was when I first started asking them.
I then say my morning prayer.
Prove it. ;)
One of the traps people occasionally fall into is that of scientism, the notion that only science can tell us that which is truthful or valuable, or that the methods of science are equally valid and applicable across all fields of inquiry. I don't think it happens all that often, but I think I would agree that Gould was occasionally prone to the affliction. It's a specific form of a general human failing - when the best tool you have is a hammer, everything around you starts looking an awful lot like a nail, if you follow my meaning.
That's not the worst of Gould's sins, though. IMO, "The Mismeasure of Man" represents one of the greatest sins in science and rationality - the distortion of the truth in order to pursue a personal political agenda. The potential damage from such poisonous perversion is immeasurable. Thanks, Steve...
I ask myself as I stand before the mirror: "How did we get here?"
Do you then tell yourself that this is not your beautiful house, and this is not your beautiful wife?
Same as it ever was, same as it ever was, same as it ever was... ;)
And more than that, the author sets up false opposites with the implication that reason can stand as equal to religion in terms of its depth, meaning and impact on our lives. Nothing could be further from the truth. Reason is a limited human tool that can deal only with what it is given. Science as well is limited insofar as it is a disciplined mode of exploration that attempts to deal with and understand what is given and what it discovers. Religion, on the other hand, goes far deeper into the source of things and, perhaps because it does, it can evoke passions that may blind believers to the importance of moral means, not ends alone. Well, I'm "preaching to the choir" here, I know. Just wanted to expand on your post.
I'm not buying what the author is selling.
Do you then tell yourself that this is not your beautiful house, and this is not your beautiful wife?
Same as it ever was, same as it ever was, same as it ever was...
You got me. Yes, I really am David Byne.
If I may add a thought here ... You read the Bible and it "touches" you. There is resonance. I have not had the same experience but I have never devoted the necessary time to its study. I have, however, been moved, and so to a degree I understand. More, I emphatically agree that this mode of knowing is far superior to cold, "objective" reason, or any of the ideologies so prevalent in the 20th Century, all of which IMHO were and are reactionary to the dominance of Christianity for so many centuries. Those ideologies resulted in rivers of blood. Islam has a far piece to go to match them. Let's hope it doesn't come to that.
The existence of God is blindingly obvious to me, and it requires tremendous intellectual effort IMHO to overcome what the vast bulk of earthly humanity acknowledges with its religions. We can pretend, for the sake of argument, that things are "out there". But where we live and where we know is "in here".
One minor example. Those of us who have been around awhile have experienced that small voice within, a gut feeling really, that has said to us "Don't do it". Well, we did, being wilful, and we paid. This has happened to me in ways small and medium large on sufficient occassions that I have begun to pay close attention to that small voice. Neither reason nor science can touch it but it's nonetheless real and true and I ignore it at my peril.
There are many many examples of experiences and truths unexplainable by science and my own view is that science and religion are false opposites. Religious knowledge encompasses science and it would be quite appropriate, again IMHO, for religious folk to say to the scientists that it is out of line when they attempt to impose restrictions on acceptable evidence (eg. replicablilty) that serve to mask science's ignorance.
I think that there is an ongoing attempt to deify "science" and "reason" here in the West which IMHO must be made to fail.
For what it's worth ...
I can't; I just believe it to be true. :^P
I wonder, what do you make of a couple of anecdotes I related on another thread recently?
Thus, he makes a false distinction between reason and religion, and he further characterizes religion (again, per se ) to mean murdering and subjugating in the name of some 'invisible spiritual entity'. Excuse me if I draw the obvious conclusion that the writer is hostile to ALL religion, not just Islamic terrorists.
As for the 'cult of rationalism', this simply describes the mentality of Randians, followers of Md. Murray O'Hare, the publishers of Skeptic magazine and many outspoken atheistic scientists who tout human reason as the highest standard in all things and often like to spell it with a capital 'R'. I do not mean that there is an organized cult or official religion, nor was it meant to be derogatory. The simple fact is that there are many people who, lacking a belief in God, substitute Reason as their god and habitually castigate all religions and all religious persons for the atrocious acts of a few extremists. However, they would be loathe to take the blame, as atheists, for all the atrocities committed by atheists throughout history (French Revolution, Marxism, Marque de Sade, etc.) Gee, THAT would be unfair, wouldn't it?
It's a certified pleasure to see you here, pariah.
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