01 Dec 05
Originally posted by ColettiPerhaps he didn't want to be redundant with Nemesio's example about stuff.
Dr. Dribble would like to avoid addressing what is simply irrefutable - that axioms are required to justify any rational knowledge - and therefore all rational world-views are founded on faith.
01 Dec 05
1 edit
Originally posted by royalchickenMathematical axioms are propositions which are not proven (i.e. they are unproven propositions - assumed true.
...Formally, all of mathematics is based on axioms, which are simply assumptions. This can be called 'faith' if you like, but with the usual connotation of the word, that sort of misses the point; mathematicians never claim to tell the truth -- they simply assert that a theorem is consistent with the axioms they have chosen. ...What is an 'unprovable axiom'?
Although you may "sort of" object to my use of the term faith, it means nothing more than something believed without proof - or assumed true. Any objection to the term is psychological, not logical.
I have never heard of a mathematician claim that 2+2 = 4 is anything but true.
01 Dec 05
Originally posted by ColettiI've never heard anyone from any walk of lie claim that it isn't. Are you going to be the first? Or are you saying that 2 may or may not be true, therefore the result of 4 is a matter of faith?
I have never heard of a mathematician claim that 2+2 = 4 is anything but true.
01 Dec 05
3 edits
Originally posted by ColettiThis is not correct. Faith, even your unusual use of it, means something more than that.
Although you may "sort of" object to my use of the term faith, it means nothing more than something believed without proof
In particular, to say that you have faith in A is to say that you believe A is true, which is to say that A is the case. But this presumes that there is the case to begin with. Only then can you have faith that A is the case.
Thus your use of the term faith entails something more than belief. It entails the existence of the case about which the belief speaks.
If you're going to claim that your use of the term doesn't presuppose the existence of the case, then you can't state that anything you have faith in is the case, which is to say that it would be absurd for you to argue that it is true. If you have faith that Jesus rose from the dead, and you think it is true that Jesus rose from the dead, then your notion of faith presumes the existence of the case of whether or not he rose from the dead. Your notion of faith is how I have described above, and not as meaning nothing more than belief without proof.
01 Dec 05
Originally posted by ColettiIf all rational world-views are founded on faith, then the term has no distinctive
Dr. Dribble would like to avoid addressing what is simply irrefutable - that axioms are required to justify any rational knowledge - and therefore all rational world-views are founded on faith.
quality. It cannot be used in any meaningful sense because it always applies and
never doesn't apply.
So, I'll ask again, is there no substantial difference between believing that Geometry
is true and believing that Christianity is true?
If there is, then what terms would you use to distinguish them? Most people would
say that belief in Geometry is not faith (even though they take the postulates
as true) and belief in Christianity is faith.
Please, give us the Colettian nomenclature that allows us to define things clearly.
Nemesio
01 Dec 05
Originally posted by ColettiYes, we do have to assume axioms, but this sort of reasoning has little to do with 'knowledge'. When I prove a theorem, I don't say 'This theorem follows from the axioms, and I take these axioms to be true, so the theorem is true.' Instead, I say 'This theorem is consistent with these axioms. Cool.' I don't care about the a priori 'truth' of the axioms, because 'true' in this context doesn't even have a sensible definition. To see this, we can have another exercise: show me a false axiom.
I humbly bow before your superior understanding of Euclidean Geometry.
Never-the-less my prior statement - "The consequence of the fifth postulate is that Euclidean Geometry can not get off the ground without presuming it is true" - is still valid.
And it is still the case that: "even mathematics is a matter of faith at some point. Not all things ...[text shortened]... d-views are founded on faith.
P.S. I did reference my quotation - so give me a little credit.
01 Dec 05
Originally posted by ColettiMathematical axioms are propositions which are not proven (i.e. they are unproven propositions - assumed true.
Although you may "sort of" object to my use of the term faith, it means nothing more than something believed without proof - or assumed true. Any objection to the term is psychological, not logical.
This is correct, but whatever source supplied that definition was a little vague, because axioms are presumed 'true' as an experiment of sorts, to see what sort of theorems can be proved from them. In what sense is an assumption true or false other than in the sense of an arbitrarily assigned truth value?
I have never heard of a mathematician claim that 2+2 = 4 is anything but true.
That's because you've only met imprecise mathematicians. What they really mean is that '2+2=4' is a theorem consistent with the axioms of arithmetic.
01 Dec 05
Originally posted by royalchickenBut you are still presupposing the axioms are correct. It does not matter what you "say" or not. If you are performing operations within the system of Euclidean Geometry - you are functioning withing a system where the axioms of Euclidean Geometry are true - because a valid proof can not follow from false premises.
Yes, we do have to assume axioms, but this sort of reasoning has little to do with 'knowledge'. When I prove a theorem, I don't say 'This theorem follows from the axioms, and I take these axioms to be true, so the theorem is true.' Instead, I say 'This theorem is consistent with these axioms. Cool.' I don't care about the a priori 'truth' ...[text shortened]... n have a sensible definition. To see this, we can have another exercise: show me a false axiom.
Axioms are necessarily true in order for your "proof" to be valid. You may ignore this fact will operating within that system - and that is fine within that system - but if you get outside that system - your proof may be invalid.
Now "natural" science operates withing the system of empiricism. Empiricism has it's axioms. To say anything is true because it is "proven by science" presuppose that the axioms of empiricism are true. However, if one does not presume the axioms of empiricism - then "natural science" proves nothing.
The presumptions of the axioms of empiricism is by definition - belief without proof. It is faith in the axioms.
Axioms are propositions that are assumed true in order for the system they infer to function. But the proposition that is by definition an axiom of it's system - is not necessarily a true proposition outside of that system. From within the system of Hyperbolic Geometry - Euclid's 5th postulate is false. It is still the same proposition - but it is not longer a true proposition.
01 Dec 05
Originally posted by royalchickenMy statement was mean to be a bit tongue-in-cheek.
[b]I have never heard of a mathematician claim that 2+2 = 4 is anything but true.
That's because you've only met imprecise mathematicians. What they really mean is that '2+2=4' is a theorem consistent with the axioms of arithmetic.[/b]
But speaking philosophically - would you say the 2+2=4 is true in reality - in the real world as we commonly know it?
Can 2+2=5 be true in any rational world-view. Or would that be an irrational statement that violates logic.
01 Dec 05
Originally posted by NemesioOriginally posted by NemesioIf all rational world-views are founded on faith, then the term has no distinctive
If all rational world-views are founded on faith, then the term has no distinctive
quality. It cannot be used in any meaningful sense because it always applies and
never doesn't apply.
So, I'll ask again, is there no substantial difference between believing that Geometry
is true and believing that Christianity is true?
If there is, then what ter ...[text shortened]...
Please, give us the Colettian nomenclature that allows us to define things clearly.
Nemesio
quality. It cannot be used in any meaningful sense because it always applies and never doesn't apply.
You are painting with too broad a brush. It applies only to all statement that are accepted as true without proof from a priori truth. It applies only to statements which are assumed true.
Originally posted by NemesioSo, I'll ask again, is there no substantial difference between believing that Geometry
is true and believing that Christianity is true?
Yes, because it involves believing different axioms. And if the axioms do not lead to logical contradiction, you may rationally believe both systems at the same time.
Originally posted by NemesioIf there is, then what terms would you use to distinguish them? Most people would
say that belief in Geometry is not faith (even though they take the postulates as true) and belief in Christianity is faith.
No new terms are required. The meaning of belief is the same, only the axioms are different. Your desire for different terms does not necessitate different terms.
It is interesting to note that you can not believe both Euclidean Geometry and Hyperbolic Geometry at the same time. You may think they are both coherent systems - but since they have contrary axioms - they can not both be assumed true at the same time.
01 Dec 05
Originally posted by DoctorScribblesFaith and belief are the synonyms.
This is not correct. Faith, even your unusual use of it, means something more than that.
In particular, to say that you have faith in A is to say that you believe A is true, which is to say that A is the case. But this presumes that there is the case to begin with. Only then can you have faith that A is the case.
Thus your use of the ...[text shortened]... faith is how I have described above, and not as meaning nothing more than belief without proof.
Faith means to believe something. Belief means to believe something. That is - to mentally assert the something is true. Most people argue that faith is belief without proof, and I'll accept that. But that means that all systems requirer faith.
You can not speak about anything if there is no context - and context is "the case". Justify that "the case" exists and does not exist and all I need to do is qualify my terms to accommodate the state of "the case".
You are dipping into existentialism and metaphysics. But I think you are not dipping in deep enough because you might drown. Go much deeper and you'll sink into non-sense.
01 Dec 05
Originally posted by ColettiBut you are still presupposing the axioms are correct.
It does not matter what you "say" or not. If you are performing operations within the system of Euclidean Geometry - you are functioning withing a system where the axioms of Euclidean Geometry are true - because a valid proof can not follow from false premises.
Axioms are necessarily true in order for your "proof" to be valid. You may ignore this f ...[text shortened]... postulate is false. It is still the same proposition - but it is not longer a true proposition.
No I'm not. The word 'correct' has no meaning at all in this sort of reasoning except 'consistent with the axioms'. I can choose whatever axioms I find interesting, and they define the notion of correctness.
In Euclidean geometry, I don't assume the axioms are 'true' other than in the trivial sense that they are consistent with themselves.
Axioms are not propositions.
