"Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned."--Avicenna

>If someone's believing a false premise is going to poison his work
>as irrational, then, of course, no one is rational. That's an awkward
>place to lead yourself.

Ah no, I never said that: Euclid thought that his axioms were immutably true, which doesn't count as a premise as we understand it.

"What is logical consistency, and why is it a necessary condition for doing mathematical work?"
Following the laws of logic: Non-contradiction and Excluded Middle (all else follows from there). It is necessary because without these, you either can do no work whatsoever, or can prove anything, depending on which way you take it. Also, there is the other fact that they are true.

There are two types of axiom, as I say again: the logical and the non-logical. In mathematics as in life you cannot break away from the logical axioms, but you can create different axiomatic systems using the non-logical axioms.

>There are, of course, further constraints on abstract speculation. One can explore an >axiomatic system for its own sake, but usually the question arises "What has this to do >with reality?" That is, we want to find an isomorphism between the results that arise in >our system (by means of applying rules of inference to our axioms in order to generate >theorems) and things that occur in the world. Sometimes the world intrudes even when >it was supposed to have been banished; recall what Gödel did to the Principia >Mathematica.

Firstly, when you ask that question, it ceases to be pure mathematics, and becomes science. Knowledge is a different concept in science, asking not "Is P true?" but "how likely is it that P is true?" You are trying to shift between two very different things all the time: between the logical and non-logical axioms and between the mathematical and scientific concepts of knowledge or truth.

What Gödel did is so often misunderstood I could tear out my own hair. He proved that: 1. Consistent sets of axioms in *arithmetic* cannot make any proposition provable. That means that you can have a proposition that is not contradictory to accept or deny. The absence of completeness is not a logical difficulty for mathematics, it doesn't make it irrational.
2. You cannot prove the consistency of a set of axioms, only its inconsistency. This means we have to be agnostic about this matter in arithmetic (note not geometry). Again, it doesn't cause any logical failure.

Number theory is a matter of logical consequences from axioms- we can do this in spite of Gödel.

>Now I'll turn to what I take to be the point of science (and mathematics is a type of >science): knowledge. Knowledge is justified true belief. Faith is belief without proof. To >make faith become knowledge, we have to add justification and truth to belief. Now, I >think you'll admit that there is at least one true thing. So if I believed that true thing, I >would have a true belief. Now, is the difference between a rational belief and faith that >the latter lacks justification? I don't think this can be definitively stated. Simply put, it is >possible for a belief to be justified without proof. If proof is not the sole means to >justification, then faith is legitimate.

Firstly, Mathematics is not science in the modern understanding. You are factually wrong there.

>How can a belief be justified without proof? Well, take Gödel's first incompleteness >theorem. This proves that there is some arithmetic statement the truth or falsity of >which cannot be proved by the axioms of arithmetic. In other words, there is NO WAY >to prove whether it is true one way or another. There are two alternatives. We could >simply suspend belief, like the ancient skeptics. We could, however, accept the truth of >the Gödel sentence without proof of it. Indeed, if we are to "take a side" regarding that >particular statement, we have to accept it without proof, because Gödel proved that >there never will be a proof available.

If something is unprovable, it means that its negation is contradictory, which means that it is consistent to accept it as true or false, and must be considered an axiom. We can prove that something is like this (take whether or not there exists and infinity between the number of reals and the number of integers as an example of this). If, however, we take a side based on faith, we will deny a mathematics that is consistent, valuable, beautiful, aesthetic, wonderful and magnificent. What's more, when you deny it you are actually wrong to do so. If you say "P is true, not P is false" rather than "P can be an axiom or not P can be an axiom", you are wrong.

However, that is anyway irrelevant, because the world is not a defined axiomatic system in the same way that mathematics is. The question of whether a religious belief is true or not is not mathematical, it is scientific: We couldn't define as axiomatic that God exists, that would be absurd, it would be an attempt of man's to definine reality. This of course renders your last paragraph obsolete.

So, to summarise: Godel did not render mathematics irrational or faith justified in mathematics, and even if it had, that is irrelevant to science.

First definition I found for science: “a branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws: 'the mathematical sciences.'”

A very unhelpful one. Science is distinct from mathematics in that it pertains to the 'real', measurable (perhaps material) world, and not to an abstract system of thought.

Logical positivism is dead; Gödel himself showed that.

What is science exactly? It's a hard thing to pin down. Is the study of history a science? I think a lot of people would say no, but I think there are elements of historical study that are scientific in nature.
I once read this definition: Science is organized knowledge. Perhaps though this is a little broad.
This is my definition: Science is a system, which is guided by reason, for discovering and organizing knowledge. This would make mathematics a science.
Was the discovery of pi a scientific discovery - as much as the double helix structure of DNA?

I can see where TGM is coming from though. There is possibly a useful distinction to be made between discoveries that pertain to real world and abstract thought. But we need an umbrella term I think. If not science, what should it be?

Acosmist, I am not a logical positivist. The fact is that you still haven't even explained what "faith" actually is.

Science in the sense that I use it can be defined as study which is based on the scientific method:

From wikipedia: "Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering observable, empirical and measurable evidence subject to specific principles of reasoning. A scientific method consists of the collection of data through observation and experimentation, and the formulation and testing of hypotheses."

This is notably distinct from mathematics, and that is why Gödel's work, even if it did what you claim, Acosmist (and I see you haven't argued against my explanation that it didn't), cannot pertain to science. My only understanding and knowledge of the world itself comes on the basis of this science: I don't even know what this 'faith' is- I've never felt it, and never had it explained to me, to do that would be the first thing, and I would appreciate it somebody would!

Faith is easy. In a nutshell, it is believing without seeing. It is accepting without physical proof. It is knowing in your heart what you rhead says you have no reason to know.

And by heart, I mean spiritual heart, your soul or spirit if you will.

"knowing in your spiritual heart, your soul or spirit"

That means nothing to be either.

But you skip the first two definitions... Interesting selective approach. I gave you two good definitions of faith that you can understand and you choose to focus on the one involving something you don't. How 13 of you, TGM.

Draugnar, if you gave two dictionary definitions, but I already know the dictionary definition of faith, but that isn't really knowing what faith is. A duck is defined in my dictionary as "a common swimming bird". You don't really know what a duck is just from the definition. "Knowing in your spiritual heart, your soul or spirit" is not a definition, however, and that is why looked to it- it was trying to communicate something I didn't understand.

To put it another way, when I ask "What is a duck?", I don't want a dictionary, I own one and could look it up. I want an explanation of what a duck is, how behaves. Similarly with faith, I want an explanation of faith, so I can come to understand what it really is; dictionaries are woefully inadequate for that.

Language is woefully inadequate then as well. For I could just as easily say "Who is TheGhostmaker?" looking for what makes you, well, you.

Using your duck example, an encyclopedia could tell you how a duck behaves and an ornithologist who specializes in Anatidae could really expand upon that, you much of what you learn could be applied to other birds as well.

Somethings have to be experienced to be understood.

A better question that is more comparable to "What is faith?" would be "What is humour?". Faith and humour are both subjective and based upon experience and the individual. Faith is what prompts me to believe. Humour is what prompts me to laugh. But, in both cases, what works for me may not work for someone else.

The beauty in nature and, believe it or not, the symmetry, order, and logic in science give me the faith to believe that God exists as random chance leads to chaos, not symmetry and order. And by extension, my faith that there is a God, allows me to feel that something more within myself (and I believe it is within the whole of the animal kingdom too, less people think I'm some kind of elitist) which leads to an acceptance that, if there is something more to me than just DNA, then that smae God who created me to send down a part of himself to teach and guide early man onto a better path that didn't involve killing animals to appease him and thinking one race is chosen and above all others (sorry, Jews, but the barbaristic practice of animal sacrifice was a pagan one to begin with and you all were no better than the satanist who do the same thing, and the attitude of being the chosen people is elitist to the Nth degree).

Applause for the faith/humor comparison Draug, that's a great way of looking at it.

But if it is subjective, and reality is objective, then how can you possibly think it valid?

Something that is subjective is clearly in some way affected by the person rather than what is outside the person. What this means is that faith surely has nothing to do with the presence or otherwise of god, and everything to do with the person who believe. It prohibits itself, by its nature, from being valid.

@vamos- that is rough....

Reality (as you put) may be objective, but perception of it is definitely subjective. Nobody has an objective view of reality. Not even scientists. Experimentation is a good way to gather data, but it is inescapable that a human scientist will have to evaluate and interpret that data. That comes down to perception of reality, not reality itself. If it were reality itself then scientists would never disagree on interpretation of the data. It tends to leave you saying which scientist is better much like a critical view of history tends to have us saying which Christians are "true Christians".

Faith is where you go when the data leaves off. Data gets you nowhere when faced with the question of God's existence. You may say that there are many other things that nobody believes in, but that isn't really an argument. It's much like saying that lot's of people agree that there is a God. It really says nothing quite eloquently. There are many ways to approach the subject, but they are all dead ends. The bottom line is that you will put your faith in something. Neither position is stronger, so it comes down to what you choose to believe.

So comedy is subjective... Does that mean there is no such thing as a comedien? Epic fail, Ghost.

Love is subjective... Should I divorce my wife because a life long commitment based on that subjective emotion can't exist?

"The beauty in nature and, believe it or not, the symmetry, order, and logic in science give me the faith to believe that God exists as random chance leads to chaos, not symmetry and order."

Assuming this is a stab at evolution as an explanation of biological diversity, lol. lololololol. Genetic mutations are random, yes. Natural selection is by definition NOT random. Organisms best suited to survive survive and pass on their traits. It's not chance, it's an individual's fitness.

It's also worth noting that comedians don't generally tell you to kill/maim the people who don't think they're funny, and people in love don't generally threaten people who tell them otherwise. In contrast, religion makes all sorts of claim, both as to holding absolute truth and how people should conduct their lives. Unapologetic superstition is ultimately responsible for the crusades, pogroms, jihads, child abuse, immolations, and all other sorts of pain and suffering. Why should anyone want to join systems with such baggage?

That's true but it can also be God's will. They are not mutually exclusive concepts. If God provided a mechanism by which his creations can adapt to their surroundings, then great it's just another blessing. Evolution, at best, criticizes one story in the Holy Bible. One story that lots of people would have agreed was probably just a creation story to begin with. My own Sunday School teacher commented that it was a creation story "nobody was on the sidelines taking notes". From my perspective, and I realize that many people theists and atheists alike, would disagree with me, this just isn't a big issue. Yes, animals adapt over time. Anyone who's been to a dog show has seen evidence of this. It is not an attack on Christianity. The real attack is in getting Christians to focus too much on one aspect of their beliefs and forget the larger picture. Living a Christian life has little to do with natural selection. To paraphrase Francis Collins, I do not believe that the creator of the Universe is in any way threatened by our attempts to understand it.

@UOSnu - It was not a stab at evolution. I believe in evolutionary theory. I just believe you can see the hand of God in it.

And my God doesn't say that either. Try reading the new testament, and not be stuck on something written by a barbaric and elitist culture like the jews had pre-Christ.

fast posted on that one :-) Mine was in answer to your first post. Next answer coming shortly.

@UOSnu

"It's also worth noting that comedians don't generally tell you to kill/maim the people who don't think they're funny, and people in love don't generally threaten people who tell them otherwise."

You're right, they generally have the good sense to leave that to the politicians.

"and people in love don't generally threaten people who tell them otherwise."

Wrong stalkers and obsessed fans do this everyday.

"In contrast, religion makes all sorts of claim, both as to holding absolute truth and how people should conduct their lives."

No crime in stating that you're right, nor in offering advice.

"Unapologetic superstition is ultimately responsible for the crusades, pogroms, jihads, child abuse, immolations, and all other sorts of pain and suffering. Why should anyone want to join systems with such baggage?"

Okay here's the crux of the debate. First of all if by "unapologetic superstition" you mean mainstream religion then that is really just innuendo designed to make your opponent look bad without having to actually argue the point. You'd be hard pressed to find anything I have to apologize for or any real evidence that my beliefs are superstitious.

Second, based on the "baggage" you lay at the feet of religion, you should be equally critical of secular governements. In fact moreso. Within the past two hundred years (especially the past one hundred) secular governents have taken the reigns with regard to atrocity. Literally millions of people died in the twentieth century due to nationalism and political ideology. This does nothing to take away from nations or politics and likewise does nothing to take away from religion. People intent on killing will find a reason to do so. If religion isn't available any other one will suffice. Remember that lots of good (charity, art, literature, philosophy, etc.) has sprung from the same tree that you would have us believe to be evil.

Crazy Anglican +1

You have been added to my list of heroes.