Irony

Originally posted by DoctorScribbles
You are utterly confused when it comes to anything regarding logic or mathematics. This is a false and nonsensical claim about Godel's theorem. I'm rather certain you don't understand the consequences of the fifth postulate being unprovable from the other four either.

The consequence of the fifth postulate is that Euclidean Geometry can not get off the ground without presuming it is true.

http://mathworld.wolfram.com/EuclidsPostulates.html

Hyperbolic Geometry assumes an alternative to the fifth postulate. And while it is a completely valid and coherent geometric system - it is radically different than Euclidean Geometry. Hyperbolic geometry holds all of the same postulates as Euclid's except the fifth.

Originally posted by Coletti
The consequence of the fifth postulate is that Euclidean Geometry can not get off the ground without presuming it is true.

http://mathworld.wolfram.com/EuclidsPostulates.html

Hyperbolic Geometry assumes an alternative to the fifth postulate. And while it is a completely valid and coherent geometric system - it is radically different than Euclidean Geometry. Hyperbolic geometry holds all of the same postulates as Euclid's except the fifth.

You're just repeating things that you have read. You have no understanding of them.

Euclidiean Geometry cannot get off the ground without presuming each and every one of the five to be true. Euclidean Geometry is the system which takes them all as axioms. There is nothing special about the fifth in this regard.

Originally posted by Nemesio
Then the term faith is meaningless because it fails to distinuish
anything about systems.

That would be like saying all extant things are made of stuff.
Q:What is stuff?
A:We can't say, but all things have it.
Q:Can we classify amongst things with stuff?
A:Yes, but not on the basis of its 'stuffness.'
Q:Then what difference does it make that ...[text shortened]... ing is exactly like
believing in the Truth of math?

If not, how is it different?

Nemesio

Nemesio! Your reproducing pre-Platonic philosophy! Good for you!

Thales (620s BC) thought that the "stuff" was water.

http://www.iep.utm.edu/t/thales.htm#H3

But he was a bit more coherent than you are here.

Originally posted by DoctorScribbles
You're just repeating things that you have read. You have no understanding of them.

Euclidiean Geometry cannot get off the ground without presuming each and every one of the five to be true. Euclidean Geometry is the system which takes them all as axioms. There is nothing special about the fifth in this regard.

You don't' understand what you read! 🙂

What is "special" about the fifth postulate is it could not be proven as a theorem.

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people.
- http://mathworld.wolfram.com/EuclidsPostulates.html

Originally posted by Coletti
You don't' understand what you read! 🙂

What is "special" about the fifth postulate is it could not be proven as a theorem.

[b]Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people.
- http://mathworld.wolfram.com/EuclidsPostulates.html

[/b]

That doesn't make it special. None of them can be proven as theorems. Show me a proof for any of them.

Originally posted by Coletti
You don't' understand what you read! 🙂

What is "special" about the fifth postulate is it could not be proven as a theorem.

[b]Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people.
- http://mathworld.wolfram.com/EuclidsPostulates.html

[/b]

Euclidean Geometry is the set of logical consequences of the five postulates we're talking about, as well as some simpler assumptions which, if we were not being careful, would be called trivial.

The fifth postulate, especially the formulation given by Euclid, is subjectively more complicated than the other four, so people wondered whether it was actually a theorem and not a postulate, and tried to deduce it from the other four.

A few people (Nikolai Lobatchevsky and Bernhard Riemann, specifically) proved that rather than being a theorem (a consequence of the other four postulates), the fifth postulate is indeed a postulate (ie completely unspecial in comparison to the others). They did this by modifying it while leaving the other four invariant, which created new geometries (sets of internally consistent consequences of the first four postulates and some fifth postulate contradicting 'the' fifth posulate).

Thus, we have that the First Four and The Fifth together give a geometry with no internal contradictions (Euclidean geometry). The First Four and ~The Fifth give a geometry with no internal contradictions (Riemann or Lobatchevsky geometry, depending on how The Fifth gets contradicted). Thus the First Four do not imply The Fifth.

Therefore, you are indeed correct that the fifth postulate cannot be deduced from the other four, but none of the other four can be deduced from any combination of the others either.

EDIT: Exercise: What's wrong with the following reasoning?

Claim: Euclid's 2nd postulate, 'Any line segment defines a unique line' is not a postulate but a theorem.

'Proof': Suppose we have a line segment PQ. Choose some point R not on PQ. By the fifth postulate, there exists a unique line through R parallel to PQ; call it L. By the fifth postulate, there exists a unique line parallel to L through P, which we call L'. Since L is parallel to PQ and L is parallel to L', L' is parallel to PQ. But L and PQ have a point P in common and are parallel, so L is the unique line containing PQ.

Originally posted by royalchicken

EDIT: Exercise: What's wrong with the following reasoning?

Claim: Euclid's 2nd postulate, 'Any line segment defines a unique line' is not a postulate but a theorem.

'Proof': Suppose we have a line segment PQ. Choose some point R not on PQ. By the fifth postulate, there exists a unique line through R parallel to PQ; call it L. By the fi ...[text shortened]... But L and PQ have a point P in common and are parallel, so L is the unique line containing PQ.

While intuitively correct, formally just about every aspect of it is flawed.

The 2nd postulate doesn't speak to uniqueness of lines.

The 5th postulate doesn't speak to existence or uniqueness of lines.

Additionally, parallel is not necessarily a transitive property under postulates 1, 3, 4 and 5.

Finally, without the 2nd postulate, the conlcusion of the 5th postulate begs the question, "Can the line segments of the antecedant be extended to be arbitrarily long?"

I imagine the latter is what you were getting at. The former are largely formal nitpicking.

EDIT: My analysis above pertains to Euclid's original formulations. I see that there are some alternative formulations for which some of my comments would not apply.

Originally posted by royalchicken
Euclidean Geometry is the set of logical consequences of the five postulates we're talking about, as well as some simpler assumptions which, if we were not being careful, would be called trivial.

The fifth postulate, especially the formulation given by Euclid, is subjectively more complicated than the other four, so people wondered whether it was ac ...[text shortened]... But L and PQ have a point P in common and are parallel, so L is the unique line containing PQ.

ahhh .... sweet memories ......

Originally posted by abejnood
We celebrate Christmas in December, but Jesus was born in March.

God created Adam, but calls Jesus his son.

God never said "Let there be sex".

About 3.2% of Christians have actually read the Bible.

About 53% of atheists have read the Bible.

Of the top ten prestigious and respected scholors on religion, 6 of them are atheists.

God promis ...[text shortened]... that everything in Islam, Christianity, and Judism is true, we're all going to hell. (Bummer).

[/b]We celebrate Christmas in December, but Jesus was born in March.

I think it was Sept., but so what?

God created Adam, but calls Jesus his son.

God created both, and they are both His sons. One is called the first Adam and Jesus is referred to as the last Adam.

God never said "Let there be sex".

Yes He did, except He intended in the marriage sense.
BTW..that's between a man and a woman.

About 3.2% of Christians have actually read the Bible.

Sad if true.

About 53% of atheists have read the Bible.

Even more sad if true.

Of the top ten prestigious and respected scholors on religion, 6 of them are atheists.

Of course they would be the most prestigious and respected if they are atheists in Satan's domain. That's to be expected.

God promises "eternal life" to those faithful, but heck, a Christian is dying every 29 seconds. (Makes you wonder).

Well...you have to die first if you want to get to heaven.

Accepting that everything in Islam, Christianity, and Judism is true, we're all going to hell. (Bummer).

I think you would have to exclude Christianity here.

Originally posted by DoctorScribbles
While intuitively correct, formally just about every aspect of it is flawed.

The 2nd postulate doesn't speak to uniqueness of lines.

The 5th postulate doesn't speak to existence or uniqueness of lines.

Additionally, parallel is not necessarily a transitive property under postulates 1, 3, 4 and 5.

Finally, without the 2nd postulate, th ...[text shortened]...

I imagine the latter is what you were getting at. The former are largely formal nitpicking.

The last part is what I was looking for. Well done!

The fifth postulate does speak to the existence and uniqueness of a specific line, in one of its (equivalent) formulations.

I took 'parallel' to be transitive because that can be proved reasonably easily. The second postulate holds that any line segment can be extended indefinitely in a line.

EDIT @ YOUR EDIT:

I should get my copy of the 'Elements' next time I go back to the States, so I can have the original formulations, and then give proofs that the versions I've used are equivalent. I mean, I could just wikipediate them, but...a maths student with no 'Elements'? For shame!

Originally posted by royalchicken
The last part is what I was looking for. Well done!

The fifth postulate does speak to the existence and uniqueness of a specific line, in one of its (equivalent) formulations.

I took 'parallel' to be transitive because that can be proved reasonably easily. The second postulate holds that any line segment can be extended indefinitely in [b]a line.[/b]

I was using the original formulations. A more challenging exercise is to demonstrate that the modern ones are equivalent to the orginal ones. I bet Coletti can show us how, as a demonstration of his mastery of the subject.

Originally posted by DoctorScribbles
I was using the original formulations. A more challenging exercise is to demonstrate that the modern ones are equivalent to the orginal ones. I bet Coletti can show us how, as a demonstration of his mastery of the subject.

Let's call it out formally:

[howl]Coletti, I gave formulations, in my exercise, of Euclid's 2nd and 5th postulates which are not verbatim quotations of those appearing in (translations of) Euclid's work. I claim that they are equivalent; it would be edifying if you would give proofs.[/howl]

Originally posted by Coletti
Yes. This includes mathematics.

Euclidean geometry is an interesting case. The "fifth postulate" was unprovable, and had to be assumed true for it to work. I think Gödel's incompleteness theorem showed that all mathematical systems have at lease one unprovable axiom.

So even mathematics is a matter of faith at some point. Not all things are a ma ...[text shortened]... ew based solely on it requiring faith - this is true for ALL world-views - including empiricism.

The only standard of truth in mathematics is consistency with the axioms. Goedel's theorems say, roughly, that for any sufficiently complex* system of axioms there exists a self-consistent theorem which cannot be deduced from the axioms; if the axiom system is sufficiently complex and can be used to prove a theorem which asserts that the system is consistent, then the theorem is inconsistent.

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. In particular, mathematics has nothing to do with empiricism, formally. Mathematics is, I suppose, a science, in one sense of the word, but it is not an instance of 'science' in the way physics and biology are.

When, as he sometimes does, my differential equations lecturer tells us that, say, a boundary condition applies to a differential equation because of physical considerations, he is no longer teaching mathematics, because in mathematics we don't care why, for instance, the gradient of some field vanishes somewhere; we just assume it. Although mathematicians historically blurred this distinction, it is no longer really acceptable or necessary.

What is an 'unprovable axiom'?

Originally posted by royalchicken


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

I think such an application of the term 'faith' is even more egregious than that.

The axioms of pure mathematical systems are just about ontologically equivalent to the rules of chess. To say that you have faith in the postulates of geometry is tantamount to saying that you have faith that the white queen ought to start on d1. That is, you believe it's conceivable that the rules have been chosen incorrectly (whatever that might mean) but you trust that they have been chosen correctly. This doesn't make any sense, for the way the universe is has no bearing on what the rules of chess should be. There is no correct or true set of rules; there only exists one within the universe that any set of rules defines. All axioms are self-fulfilling prophecies by nature - there is no need for faith in them.

Euclid could have formulated quite different postulates, and Lobachevsky actually did! Must one be right and one be wrong? No, because they don't assert truths about the universe. Neither states that anything is the case in our universe. Rather, they define their own universes and then serve as fodder for theorems which assert truths about that universe.

The power of mathematics lies in its standard of proof, not in its standard of truth.

Originally posted by royalchicken
Let's call it out formally:

[howl]Coletti, I gave formulations, in my exercise, of Euclid's 2nd and 5th postulates which are not verbatim quotations of those appearing in (translations of) Euclid's work. I claim that they are equivalent; it would be edifying if you would give proofs.[/howl]

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 are a matter of faith, but all systems depend on the assumption of something unprovable" which is my main point.

The fact that E.C. requires the assumption of all 5 postulates only goes to reinforce my position that we can not know anything - not even mathematics - without assuming something that can not be proven a priori.

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.

P.S. I did reference my quotation - so give me a little credit.