Social Question

RealEyesRealizeRealLies's avatar

What is Proof?

Asked by RealEyesRealizeRealLies (30938points) October 2nd, 2009

Does Proof equal Truth? Or is Proof the result of a proposition being True Enough? Is there such a thing a absolute Proof?

Fluther knows when my computer logs on because a distinct I.P. address is associated with that task. Is that data considered absolute Proof?

If a video tape surfaced that showed Christ rising from the dead, is that considered absolute Proof? Would that confirm the supposed eye witness accounts?

My child spilled his milk right in front of my eyes this morning. Is there any Proof that he actually did that?

Is Proof always a form of data? Or can it be pure experiential in nature? Does a thing called Proof actually exist? Is it good enough?

Does Science set out to Prove a thing, or to disprove all other possibilities instead?

If God herself looked you in the eye and said, “I AM”... Is that proof of her existence?

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85 Answers

Firstandlast's avatar

Proof is what we can show is real. Someone can make any claim no matter how bizarre but it is the proof that makes it real to other people. If you met someone who didn’t know a thing about dinosaurs and you drew them a picture they would probably not take your word for it but if you took them to a museum and they saw the proof of their bones. They could never dispute that yes dinosaurs once lived on earth.

se_ven's avatar

seems to be one of those questions where the more you think about it the more confused you make yourself…

RealEyesRealizeRealLies's avatar

@Firstandlast
I think what you speak of is evidence, not proof. Yes, there is evidence that large reptilian creatures once walked the earth. But how do you know they weren’t criminals from an intelligent and highly advanced ancient extraterrestrial giant lizard planet, stripped naked and dumped off here to be executed by the asteroid that soon followed?

Evidence is one thing. Is speculation about that evidence to be considered as absolute proof?

virtualist's avatar

The axioms of logic and classical logical analysis are absolutely true….. from that one can create all of mathematics as we know it….... Godel would remind us tho’ that we cannot prove that our analysis is ‘complete’.......

Firstandlast's avatar

@RealEyesRealizeRealLies, yes, I think you make a valid point. Thanks

RealEyesRealizeRealLies's avatar

@virtualist
I’m so very happy you said “create” the mathematics. It’s unfortunate that many people believe we “discover” them. We don’t.

gussnarp's avatar

What constitutes proof depends on what is being proven and to whom. Sure, one could argue that there is not absolute proof, but if that is the case, what good is the word? Bones are evidence of the existence of dinosaurs, and for most people the fact that this evidence has been found repeatedly all over the world is proof that dinosaurs existed. No, it does not prove where they came from, but it is sufficient proof for reasonable people to be willing to assume they existed. Proof (outside of mathematics) is sufficient evidence to convince a particular audience. In a court of law that means to convince twelve individuals beyond a “reasonable doubt”. That reasonable doubt, I would imagine, is defined within the mind of each juror.

RealEyesRealizeRealLies's avatar

@gussnarp
True enough, but you well know of many individuals who were wrongly convicted from evidence considered as proof beyond a reasonable doubt.

I propose that Proof is that which predetermines the evidence, not the other way around.

se_ven's avatar

@RealEyesRealizeRealLies you’d make a good defense lawyer ;)

gussnarp's avatar

@RealEyesRealizeRealLies Do you know me? On point one I couldn’t agree more, on point two: Huh?

virtualist's avatar

@RealEyesRealizeRealLies Actually, I should have said that all of mathematics ‘evolves’ from that perfect starting point of ‘pure logic’.

eponymoushipster's avatar

It’s what a Chinese magician says right before he pulls a rabbit out of his hat.

RealEyesRealizeRealLies's avatar

@gussnarp
Point 2 – The first question in my details asked if Proof was equal to Truth. I believe that it is. Proof is the Truth we can know, thus it predetermines the evidence.

Firstandlast's avatar

@gussnarp – Thanks for understanding my dinosaur reference.

RealEyesRealizeRealLies's avatar

@virtualist
And what is the genesis of ‘pure logic’...?

gussnarp's avatar

I don’t think proof is synonymous with truth. Proof is how we judge truth, it is not the truth itself.

RealEyesRealizeRealLies's avatar

Evidence is judged. Proof is accepted or denied.

gussnarp's avatar

I like this definition from Webster’s: the cogency of evidence that compels acceptance by the mind of a truth or a fact. Webster’s further defines evidence as an outward sign or indication. I think proof can be better conflated with evidence than with truth. Both evidence and proof are external to truth, they are signals of truth, not truth in and of themselves.

The_Compassionate_Heretic's avatar

Proof is factual evidence that helps establish the truth of something.
-wordnetweb.Princeton.edu

RealEyesRealizeRealLies's avatar

@gussnarp
Worth considering. I don’t like that phrase “compels acceptance”.

Evidence can be valid or invalid. Evidence can be manufactured. If proof is “the cogency of evidence”, then proof could be the result of manufactured lies. I have a big problem with that.

Again I claim, that proof is the Truth we can know.

RealEyesRealizeRealLies's avatar

@The_Compassionate_Heretic
That comes closer. The evidence must be factual. So a tautology develops.

Fact – Evidence – Proof – Truth

gussnarp's avatar

I don’t think that’s a tautology. I think that evidence is not evidence if it is invalid or non-factual, but if we feel the need to clarify that the evidence is factual, then I am happy with that addition to the definition. As to “compels acceptance” I think that is the heart of what proof is. Proof is not truth, it is what is sufficient to induce one to believe something is true. Proof is relative. Then again, so is truth.

RealEyesRealizeRealLies's avatar

@gussnarp said:
“Proof is relative. Then again, so is truth”.

Did you just make an absolute statement about a relative supposition?

The_Compassionate_Heretic's avatar

It’s important to note that proof is objective whereas our perceptions of what we think proof is, is a subjective thing.

Haleth's avatar

There’s a bar where I live called Proof. I was kind of hoping this question would be about… something like that.

LostInParadise's avatar

Nothing, absolutely nothing, is known for certain. That is the assumption of science. We can only tentatively believe something, because all attempts so far to refute it have failed. What can be more certain that if you are going 60 mph and someone approaches you going 40 mph that you pass each other at 100 mph? Newtonian physics says it is so, but relativity says that it is not quite accurate.

Having made that statement, we have to live our lives on the assumption that we can believe certain things. If we see a child spill milk then we assume this to be so. We also have to go along with the assumption that we are able to accurately apply logic.

Mathematics is a special case. Mathematics says nothing about the real world. It just says that if certain axioms are assumed then certain results follow. Mathematics says that 1 + 1 = 2, but it does not say that one apple plus one apple equals two apples, because it has nothing to say about apples.

gussnarp's avatar

Should I have said proof is relatively relative?

RealEyesRealizeRealLies's avatar

@gussnarp

Courts consider all evidence. Some is rejected as invalid for whatever reason. But it was presented as evidence nonetheless. There are two types of evidence, valid and invalid.

RealEyesRealizeRealLies's avatar

@LostInParadise said:
“Nothing, absolutely nothing, is known for certain”

How can we know such a thing?

The_Compassionate_Heretic's avatar

@LostInParadise That statement is an absolute and absolutes are generally flawed statements.

RealEyesRealizeRealLies's avatar

@gussnarp

There is absolute proof that the statement “Should I have said proof is relatively relative?” appears four threads above this one.

gussnarp's avatar

Or is it five?

RealEyesRealizeRealLies's avatar

@gussnarp

As @virtualist says,
”...all of mathematics ‘evolves’ from that perfect starting point of ‘pure logic’.”

virtualist's avatar

…......We can all now go happily away and study Popper, Kuhn, Sokal, Feyerabend, and Lakatos and reconvene later in the year to answer @RealEyesRealizeRealLies last question:

”...If God herself looked you in the eye and said, “I AM”... Is that proof of her existence?”

My naive answer , before studying up on all this would be that…..”.. you started with the ‘conclusion’ , thus it is not scientifically verifiable or falsifiable…..... thus I am God !”

Axemusica's avatar

Proof and that’s all the proof I need. Having the evidence is proof. Providing the evidence is proof. Asking for proof is asking for the evidence. Simply words.

If you don’t have proof of insurance when the police officer pulls you over, it’s evident that you might be getting a hefty fine.

virtualist's avatar

@RealEyesRealizeRealLies ....... going back… the answer to your question regarding the ‘genesis of pure logic’ ......... that would be Aristotle, as documented in his 6 book collection called the ‘Organon’......

LostInParadise's avatar

Everything is uncertain because there is no way of determining certainty. Everything known about the world is empirically determined. The best that can be done is to perform an experiment, and just because you get the same experimental results 999 times in a row does not guarantee that the 1000th time will be the same.

YARNLADY's avatar

As far as the proof of the spilled milk, the question becomes is the world you percieve even real? What is the proof.

If having to clean up the milk constitutes proof for all practical purposes, that is all you need, but if you want a physical/philosophical proof, there is none.

Ivan's avatar

Proof does not exist in the natural world. All we can do is make sure that our theories explain what they intend to explain within the context of the known evidence.

virtualist's avatar

Mathematical proofs exist in the real world…............. in any world…......... forever and ever !!

If you want to communicate with ‘aliens’ ......... start with logic and math… like the original voyager ...... Voyager 1 – Math Msg -

RealEyesRealizeRealLies's avatar

@Axemusica

The reason I asked this question is because I disagree with the standard definitions of Proof. It’s a circle jerk of words that compliment each other but don’t explain a thing.

In particular, I take great issue with dictionary defs (like the first one you provided) that use this type of phrasing:

“evidence sufficient to establish a thing as true…or to produce belief in its truth.”

That is psychosis, not proof. By those standards, Jim Jones, Marshall Applewhite and David Koresh have visions that they consider as “sufficient evidence to establish a thing as true”.

Balderdash.

If you had a video tape of me stabbing a victim to death, did that prove anything? She actually died from a brain tumor instead, and would have otherwise survived my assault.

RealEyesRealizeRealLies's avatar

@virtualist

If that be the case, and the Organon is the genesis of pure logic, would not Aristotle have needed to produce his thesis about it without the use of logic?

What is the ultimate genesis of the pure logic that you refer to?

Axemusica's avatar

@RealEyesRealizeRealLies well, in that case I guess proof would be the evidence that shows the most logical speculation. So therefor some people would not agree that “stabbing a victim to death, that died from a brain tumor” would be considered murder.

The weight of the stone only matters to that who it is being held by.

RealEyesRealizeRealLies's avatar

@virtualist

That is not a mathematical proof. That is a codified description of a theoretical mathematical entity. The entity is not the same as the description. I can describe the entity many different ways beyond the Voyager message, but I can never prove the entity actually exists because it is comprised of pure immaterial information. The entity is immaterial. The description of it is material. They are different things altogether. One represents the other, the other which cannot be proven to exist.

RealEyesRealizeRealLies's avatar

@Axemusica

In a round about fashion, that somewhat hints at what @Ivan said. However your position still places the burden upon the subjective interpretation of what a “logical speculation” actually is. If Proof actually exists, shouldn’t it be more objective in nature, independent of personal bias?

Axemusica's avatar

@RealEyesRealizeRealLies “If Proof actually exists, shouldn’t it be more objective in nature, independent of personal bias?” Who’s to say nature exists? Objectivity is also in the eye of the beholder. Things are as we perceive them to be. To be completely accurate and “true” we’d need an absolute outside source, in which you answered to @virtualist is “immaterial.” Will the answer succumb? Who knows, but we can agree that this is truely an interesting debate, lol.

virtualist's avatar

@RealEyesRealizeRealLies #1— re: Organon

Popper’s answer to the question of ‘which came first, the chicken or the egg’ was ’ ... a prior egg’ ......... Aristotle codified formal logic in his mind… an original act of genius ! ... and I’m not saying it only took him 20minutes…........ he thought about it deeply and codified a ‘way of thinking’ ...... which is fundamental beyond anything I , personally, imagine.

RealEyesRealizeRealLies's avatar

@Axemusica
“Who’s to say nature exists?”

Nature is natural. Natural is that which is possible.
Hence, if anything is possible, then nature exists.

“Objectivity is in the eye of the beholder”

Absolutely

“Things are as we perceive them to be”

Things are things, completely independent from our perception of them. Otherwise I didn’t exist until your perception brought me to life. Thanks, I think.

virtualist's avatar

@RealEyesRealizeRealLies #2 – No where did I say they communicated a proof….. they sent the basis for basic maths and codified / symbolized that…........ in an intriguing UNIVERSAL manner that could potentially speak to an intelligent silicon vegetable based ET !

RealEyesRealizeRealLies's avatar

@virtualist
“chicken or egg”...?

Neither can occur without prior codified information to determine them in advance. The answer to the question is Information came first.

And what have we here? Aristotle “codified” formal logic in his “mind”

Interesting association that codified information comes from a mind. Hence the genesis of logic is mind… yes?

RealEyesRealizeRealLies's avatar

@virtualist
#2… I did not mean to misrepresent your words. Unintentional.

Axemusica's avatar

“Nature is natural. Natural is that which is possible.
Hence, if anything is possible, then nature exists.”

If this is the case than proof is definite according to Super String Thoery and who’s to say that proof isn’t just that in another dimension? How would one find the answer to that question anyway? lol.

Although, nature might not be nature to another dimension and to that dimension the natural might be absent of nature. What evidence do I have to provide you with proof of this theory? None, but there’s no evidence that proves otherwise. The question of “proof being absolute truth or not” is a conundrum if you don’t include the fact that thought it’s self is bias and thus a natural part of life.

“Things are things, completely independent from our perception of them. Otherwise I didn’t exist until your perception brought me to life. Thanks, I think.” That’s your perception and your welcome ;)

RealEyesRealizeRealLies's avatar

@virtualist

What’s interesting is that both Vogager and SETI base their search for life upon codified information. Some scientists want us to believe that life is a matter of “just add water”.

Life = Energy + Matter + Information

How’s that for a fundamental equation?

RealEyesRealizeRealLies's avatar

@Axemusica

Evidence can be discovered, manufactured, presented, and considered. But it can never prove anything.

Axemusica's avatar

@RealEyesRealizeRealLies True. It also doesn’t disprove anything. Some might disagree that using tires on vehicles is proof that it gives you more traction. Some might also disagree that plucking a string on a guitar proves that it makes sound, but it doesn’t mean those things aren’t actually happening, then again according to theory I’m not sure what it’s called that choice does not exist, then all those things have already happened and this debate is already over.

I can see you’re very objective and I think it’s a good thing and in fact I’m joining your fluther, hehe. :) plus lurve!

RealEyesRealizeRealLies's avatar

@YARNLADY
”...the question becomes is the world you percieve even real?”

That is the question. I prefer to think the world is the world, regardless of my perception of it. My perception of it is what Robert Anton Wilson calls the “Reality Tunnel”. We all have our own Reality Tunnel. But that in no way disregards the existence of an actual objective world around us, no matter what that may be like.

Did the “personhood” of my son spill the milk, or were there other factors that contributed to cause/reaction that spilled it? Lighting, muscle spasm, grogginess… None of those are part of the personhood of my son. So did my son really spill the milk, or was it simply a chain of events brought about by cause and reaction? No thought was present, so how could he have spilled it?

Yet the evidence suggests that milk was indeed spilled nonetheless.

RealEyesRealizeRealLies's avatar

@Axemusica

Nitpicking for fun…
Plucking a guitar string does not create a sound. It creates an atmospheric disturbance in the air molecules in the form of a wave. Sound occurs when the eardrum transduces that vibration into an impulse that the brain can perceive.

Sound depends upon a brain to call it that. Everything else is simple vibration.

RealEyesRealizeRealLies's avatar

@LostInParadise
It does seem that way. But it doesn’t really seem that way. But it sure seems to be that way.

Axemusica's avatar

@RealEyesRealizeRealLies “Everything else is simple vibration.” Super string theory :P

Although I did mean in lamens terms. Nonetheless yes, plucking the string is an action with a whole string of reactions. :P

RealEyesRealizeRealLies's avatar

@Axemusica

So you like string theory? Do you have any issues with it? I think it’s interesting.

Axemusica's avatar

@RealEyesRealizeRealLies I love it. There’s a book I read that turned me onto it. Sex, Drugs, Einstein & Elves. Great book. I think you’d like it. ;)

dannyc's avatar

Proof is what leads you to your conclusion of truth. Usually to be superceded by new proof and your changing beliefs.

YARNLADY's avatar

Who spilled the milk? The dog did it.

virtualist's avatar

@all

I’ve been away for 2.5002 hours and I have no chance of catching up…......
@Axemusica…. I’ve definitely gotta start reading Pickover’s books

@RealEyesRealizeRealLies…“Life = Energy + Matter + Information”... but E and m are equivalent AND E and I are each conserved THUS Life is conserved, MEANING either there is L after death OR we are all immortal. Also, the answer to all your ?‘s is ‘YES’ ..... well, ... ok…..... “The dog did it.” answers one of them, per @YARNLADY.

What a hoot…. this has been !

whatthefluther's avatar

@RealEyesRealizeRealLies….What a delight to see you back after an apparent hiatus. You are back, are you not? Can you prove that? Interesting question that begs a bit more sobriety than that which I am currently in a position to offer. With that in mind, I can offer this: the proof is in the pudding and the truth is, I prefer tapioca. Again, welcome back. See ya later…Gary/wtf

Axemusica's avatar

@whatthefluther lol

@virtualist After that book I really want to get a few other books of his. I just haven’t gotten around to it. That book is a great start, I tried reading the first time got half way through and become real busy, came back to it read it all and I’m thinking about reading it again. The section on DMT is by far my favorite.

RealEyesRealizeRealLies's avatar

@virtualist

Yes E and M are essentially the same. And although both are conserved, what of @virtualist‘s essence remains in that conservation? Becoming worm food is not my idea of “L after death” OR “immortality”.

Only the agent of Information is immaterial. We are beings based primarily upon a finite quantity of Information. We are created from Info, and we also continually craft our essence during physical life by authoring additional Info as we live.

In the beginning was the Word. And the Word was God and the Word was with God. And the Word became flesh. What is the fate of the Word after the flesh becomes grubdung?

RealEyesRealizeRealLies's avatar

@whatthefluther

I pictured you more as the Creme Brulee type.

virtualist's avatar

@RealEyesRealizeRealLies There is no information in the Word. Thus there is nothing to be conserved at grubdungung. As BRussell said ’...the burden of proof is on the believers’.

RealEyesRealizeRealLies's avatar

@virtualist
Correct, there is no information in the Word. Words only represent Information. They are different things. Words are material. Information is immaterial. It’s actually “proof” of an immaterial realm.

“Information is Information. Not Energy and not Matter. Any materialism that does not allow for this cannot survive in the present”.
Norbert Weiner, Cybernetics p147

mattbrowne's avatar

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception (Wikipedia).

I love proofs by contradiction. For example the supposed rational number sqrt(2) = a/b where a and b are non-zero integers

LostInParadise's avatar

But mathematics by itself tells us nothing about anything outside of matematics

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Bertrand Russell

RealEyesRealizeRealLies's avatar

Matt, am I incorrect in that mathematics is nothing more than a description of a theoretical entity? But it is not the entity, it only describes one.

In that case, the theoretical entity is completely immaterial. We cannot prove that it actually exists. We can only prove the material description, but not the immaterial entity.

For example, I have material proof that:
“sqrt(2) = a/b where a and b are non-zero integers”
was written as a description two threads above this one.

But I have no material proof that the immaterial entity it represents is actually real.

LostInParadise's avatar

I don’t mean to pre-empt matt on this, but you are absolutely correct. The physicist Eugene Wigner has a great phrase to describe the situation. He talks about “the unreasonable effectiveness of mathematics.” When mathematicians talk about undefined terms, they are not saying that we know what these things are but we just can’t define them. Unknown terms means completely abstract entities that are constrained only by the associated axioms relating them.

What really brought this home originally was the discovery of non-Euclidean geometry, which had no impact on the man in the street, but led to a final conceptual separation between math and science. By jettisoning the parallel postulate, which everyone was kind of uneasy with, you could get a geometry very different from Euclidean geometry and nobody knows for sure what the geometry of the Universe is, since non-Euclidean geometry is locally Euclidean.

RealEyesRealizeRealLies's avatar

@Axemusica

I’ll have to get that book… Huge McKenna fan. No one else like Terrence. Is it written by him or does it just review his theories?

Axemusica's avatar

Huge McKenna? Dunno who that is. The book I speak of is a Clifford A. Pickover book & yes it’s written by him. So much in fact sometimes you feel like you’re actually having a conversation with him on a park bench, he’s just doing most of the talking, lol.

RealEyesRealizeRealLies's avatar

@LostInParadise

Having trouble grasping some of your comment. What do you mean by:
“they are not saying that we know what these things are…”?
”...but we just can’t define them”

Is that the same as saying
“they are saying they don’t know what these things are”?
and so they cannot define them?

I think I’m understanding you, but just trying to make sure. It seems to me that even an abstract entity should have a description… perhaps I’ll have to settle for a phenomenal label like “abstract entity”, but how is it theorized upon without a description?

And so what is meant by “Unknown terms”?
“Unknown terms means completely abstract entities…”

I fail to see the logic in using another phenomenal label “Unknown terms” to describe the original phenomenal label “abstract entities”.

@LostInParadise said:
“the discovery of non-Euclidean geometry… led to a final conceptual separation between math and science”

That doesn’t surprise me at all. Mathematics is after all, a language, and much more akin to Linguistics. Science typically adheres to strict principles of hard materialism. Mathematics and Linguistics can break free of those bonds and explore the immaterial realm of abstract symbolism.

RealEyesRealizeRealLies's avatar

@Axemusica

I see. McKenna speaks of Pickover on occasion. The “Elve’s” are out of McKenna’s head though. And he has much to share about DMT, Psilocybin and 2012.

Find McKenna. Do you deoxy?
http://deoxy.org/

Axemusica's avatar

@RealEyesRealizeRealLies “The Elve’s” are out of many peoples heads. Most of the experimentation done using DMT and of people using psychedelics reported “elve” like beings as if constantly working on “the strings of life” Superstring Theory ;), I myself have hallucinated not with DMT, but I want to, and I did feel as though there was something more than just what we are in “life.” It’s kind of hard to explain, all I can say is if you haven’t “tripped” you’re missed out on some life experiences.

Though, now that you say that it does sounds familiar. interesting site. I’m gonna have to check that out later. I’m a bit tired right now, lol.

RealEyesRealizeRealLies's avatar

@Axemusica

I truly believe McKenna was the first to coin the descriptive label “Self Dribbling Elf Machines” in his book The Invisible Landscape and it is a huge subject in most of his lectures as far back as Tim Leary days. There are some who speculate that others who see them are actually under McKenna’s subconsciously planting the image in their mind before they ever went there.

I don’t know personally. But I will.

mattbrowne's avatar

@RealEyesRealizeRealLies and @LostInParadise – I think your concerns are very valid and they relate to the much deeper philosophical question: Is mathematics independent of physics?

I’m reading David Deutsch’s book “Fabric of Reality” right now and he’s touching this subject. There’s also some material on the Internet, including a highly interesting discussion which took place in 2006 at the University of Auckland:

Cristian Calude: I suggest we discuss the question, Is mathematics independent of physics?
Gregory Chaitin: Okay.
CC: Let’s recall David Deutsch’s 1982 statement:

The reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic, cannot be found in mathematics or logic. The reason is that the laws of physics ``happen” to permit the existence of physical models for the operations of arithmetic such as addition, subtraction and multiplication.

Does this apply to mathematics too?

GC: Yeah sure, and if there is real randomness in the world then Monte Carlo algorithms can work, otherwise we are fooling ourselves.
CC: So, if experimental mathematics is accepted as ``mathematics,’’ it seems that we have to agree that mathematics depends ``to some extent’’ on the laws of physics.
GC: You mean math conjectures based on extensive computations, which of course depend on the laws of physics since computers are physical devices?

See the whole discussion:

http://www.cs.auckland.ac.nz/~chaitin/md.html

LostInParadise's avatar

Matt, That is a good point about experimental mathematics. Another example of the use of the real world to do mathematics is the use of a computer program to prove the four color theorem.

@RealEyesRealizeRealLies , Let me give an example of what I mean by abstract entities. When we think of a line, we have a definite picture in mind, but the mathematician would say that a lines and points are anything that satisfies the axioms, like two points determining one line. For example, the legitimacy of non-Euclidean geometry was shown through the use of Euclidean geometry. I don’t know the details, but the basic idea was to look at a space limited to the interior of a circle and to define lines to be a certain type of circular arc. It was then possible to use Euclidean geometry to show that these circular arcs satisfied all the line axioms except for the parallel axiom.

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