Are axioms beliefs?

[T'ai Chi]

Axioms might be beliefs that everyone considers self evident, but doesn't that still make them beliefs?

 

[Iacchus]

Axioms might be beliefs that everyone considers self evident, but doesn't that still make them beliefs?

I would yes, because the truth is not always self-evident, that is until one "discovers" it for oneself. In which case there may be a lot of people who believe such things are so but, they don't actually know.

Also, there may be some people who, for one reason or another, have no ability to see it ... Not everyone can ascertain whether 1 + 1 = 2.

 

[Yahweh]

Everything is a belief, its the inherent truthfulness of the beliefs which I'm concerned with. However, I dont think its correct usage of terminology to say the word "axiom" is equivelant to the word "belief".

Axioms are defined as "in logic: a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident". Usually definitions like this are misrepresented (such as if someone says "Its self evident that God exists"), I recommend reading Philosopher G.E. Moore's "A defense of Commonsense" for a good idea of "Things which I know are true but cannot prove absolutely".

These are all axioms:
* The external world exists
* The contents of a book do not change after you close it
* Things which have a front side also have a back side
* The gravity of Earth will not suddenly invert on me
* Humans wont be morphing into cats within the near future if ever
* In Physics, most proofs assume the mass (or other such variables) of system will remain constant (this is also helpful to fend off a series of "what if" questions)

 

[Soapy Sam]

Axioms are assumptions which act as a foundation for constructs , logical or illogical.
Noone has to believe them, though that's an individual decision.* They are taken as correct for the purpose at hand and are frequently restricted to special cases. They may not apply to the world in general.



* Actually, I wonder . Do we have any choice about what we believe? I have no idea why I never believed what I was told in Sunday School. I suspect I just didn't like the teacher and so rejected the teaching. Doomed to eternal perdition because of a lady with a wart on her lip. Life is hell.

 

[MRC_Hans]

In principle, axioms do not equal beliefs. An axiom is whatever is agreed upon as a basis for a given discussion. The axiom itself might be the subject of another discussion. That said, the axioms for most discussions do coincide with commenly held beliefs.

Yahvee, I'm sure you can find, even on this board, people who are not ready to accept several of the axioms on your list. Now, our wee Franko has left for parts unknown, but otherwise you might have had a long debate with him on #1 (and you might still try it on II ).

Hans

 

[slimshady2357]

Now, our wee Franko has left for parts unknown, but otherwise you might have had a long debate with him on #1 (and you might still try it on II ).

Hans

Not unless he meant mind-idependent external reality

originally posted by Interesting Ian
I have said no such thing. I have never denied an external reality.

Said here

Adam

 

[xouper]

T'ai Chi: Axioms might be beliefs that everyone considers self evident, but doesn't that still make them beliefs?

Just to throw a monkey wrench into the conversation, axioms don't have to be self-evident. Consider for example the mutually exclusive alternatives to Euclid's Fifth Postulate, none of which appear to be falsifiable.

 

[T'ai Chi]

I'm still waiting for some custodian, plumber, or some other expert in mathematics to chime in and correct me after making fun of me about my statement that axioms are beliefs.

Where oh where can he be??

Typical.

 

[hammegk]

Re: Re: Are axioms beliefs?

Just to throw a monkey wrench into the conversation, axioms don't have to be self-evident. Consider for example the mutually exclusive alternatives to Euclid's Fifth Postulate, none of which appear to be falsifiable.

Well, Yahweh did say "assumed to be self-evident", not "are self-evident". What is "self-evident" to you?


Yahweh: On your axiom list, you might do some reading/thinking on, for example, a priori/a posteriori.

 

[geni]

These are all axioms:
* The contents of a book do not change after you close it
Only at 0K (and maybe not even then)
* Things which have a front side also have a back side
I'm sure if you mess with space time enough you can find a way round this

Sorry

 

[Zombified]

Axioms are neither 'beliefs' nor necessarily 'self-evident.'

The classic example is Euclid's axiom regarding parallel lines (that they never converge but remain a constant distance apart). This axiom only applies when you are talking about a certain class of geometries, such as planar geometry (a flat 2D plane).

You can replace this axiom with a different version and get different classes of geometries. For example, there's geometry on the surface of a sphere, where lines are defined to be great circle (that is, the center of the circle is coincident with the center of the sphere you've defined your geometry on). In this geometry, all lines eventually cross.

Both sets of axioms are true in all the cases where they apply (and belief does not enter into it). They are also mutually exclusive.

I prefer to think of axioms as a definition. Axioms define a class, and anything that behaves according to those axioms is an instance of that class. Any conclusion proven by the axioms can be drawn about a particular instance. Euclid's axioms of geometry define what it means to be a Euclidean space. The axioms of spherical geometry define the class of spherical geometries. The axioms of the natural numbers define the natural numbers. Etc.

To say that a set of axioms is "true" is to make a statement about the applicability of a set of axioms to a situation. If you "believe" Euclid's axioms, you are basically asserting that you believe space follows those rules.

 

[xouper]

Yahweh: * Things which have a front side also have a back side

geni: I'm sure if you mess with space time enough you can find a way round this

You mean like a mobius strip, which has only one side and one edge?

http://www.physlink.com/Education/AskExperts/ae401.cfm

 

[Captain Trips]

These are all axioms:
* The external world exists

Nope. It goes away when I close my eyes.

* The contents of a book do not change after you close it

What about throwing it in a fire without reopening it? THEN it will change.

* Things which have a front side also have a back side

As has been pointed out, just look at a Moebius strip

* The gravity of Earth will not suddenly invert on me

Unless a black hole appears overhead.

* Humans wont be morphing into cats within the near future if ever

Seen the latest Harry Potter movie?

* In Physics, most proofs assume the mass (or other such variables) of system will remain constant (this is also helpful to fend off a series of "what if" questions)

In quantum mechanics, nothing is constant. In fact, nothing "is" -- it is all probability.


Just to throw a monkey wrench in the works. I would think that axioms are "agreed upon", but given a different set of assumptions these "truths" fall apart. Since they rely on certain assumtions, I would tend to consider them as "beliefs" as you suggested.

Of course, that's just my opinion. I could be wrong, in which case I have made an incorrect assumption -- I believe.

 

[Darwin'sGoat]

Where oh where can he be??

You sir should change your name to Devil's Advocate. I have checked, it is not taken.

Good old dictionary.com tells me that an axiom is:

1. A self-evident or universally recognized truth; a maxim
2. An established rule, principle, or law.
3. A self-evident principle or one that is accepted as true without proof as the basis for argument

Belief is listed as:

1. The mental act, condition, or habit of placing trust or confidence in another
2. Mental acceptance of and conviction in the truth, actuality, or validity of something
3. Something believed or accepted as true, especially a particular tenet or a body of tenets accepted by a group of persons

An assumption, the inevitable third word you like to toss out like a linguistic landmine is:

1. The act of taking to or upon oneself
2. The act of taking possession or asserting a claim
3. The act of taking for granted: assumption of a false theory
4. Something taken for granted or accepted as true without proof; a supposition: a valid assumption
5. Presumption; arrogance
6. Logic. A minor premise

Is an axiom a belief? Yes an axiom is a belief. A belief in something that is self-evident. It often seems that you are threatening to argue things from a Nihilistic viewpoint. Could you please just come out and say it if you are?

Your argument will undoubtedly be: "Since an axiom is a 'proposition that is not susceptible of proof or disproof'" how does it differ from someone's religous beliefs which can be neither proved nor disproved.

My argument to this statement (were you or anyone else to make it) would be that the deductions, assumptions and proofs that follow from the scientific axioms continue to explain, in exacting detail, the world around us. Whereas the deductions, assumptions and proofs that follow from the religous axioms crumble under their own weight.

The exciting part of our discussion however centers around your insistance that the thesaurus proves that:

an axiom = a belief = an assumption

therefore,

an axiom = 3. The act of taking for granted: assumption of a false theory

and

an axiom = 5. Presumption; arrogance

This 'mathematics' as you call it is faulty because a thesaurus does not prove that those things are equal. It proves that those things are similar.

1. Related in appearance or nature; alike though not identical

I had assumed that my (oh so clever) 'proof' where I used your own logic to show that 'Tai Chi = Endless Poop' illustrated this beautifully. It makes me sad to think that my uncanny wit was wasted. It was, like so many other good points that people make against you, blissfully ignored. Instead you cried 'impugn' and stormed off.

You closed by telling me to "find the root of the word 'axiom'" Following the helpful link you provided I discovered that the root of said word is from the Greek, axios, which means worthy.

My steely ego affords me no other alternative than to accept your humble peace offering.

Thank you.

 

[Darwin'sGoat]

I would also like to add, that just because an axiom is a belief, that does not in turn mean that beliefs are auto-magically axioms.

Just wanted to add that one in case anyone was planning on making the assumption.

 

[Zombified]

Good old dictionary.com tells me that an axiom is:

1. A self-evident or universally recognized truth; a maxim
2. An established rule, principle, or law.
3. A self-evident principle or one that is accepted as true without proof as the basis for argument

Those are colloquial usages...

 

[Darwin'sGoat]

Yeah, my original statement was "Yes, according to those definitions, an axiom is a belief." But I thought is sounded evasive. In retrospect I wish I had left it.

But, I am trying to argue the term. Granted the argument keeps blurring between the colloquial usage and the technical, but...

 

[Darwin'sGoat]

Re: Re: Are axioms beliefs?

Also, there may be some people who, for one reason or another, have no ability to see it ... Not everyone can ascertain whether 1 + 1 = 2.

And that would make it untrue for what reason?

 

[Zombified]

Re: Re: Are axioms beliefs?

I would yes, because the truth is not always self-evident, that is until one "discovers" it for oneself. In which case there may be a lot of people who believe such things are so but, they don't actually know.

Also, there may be some people who, for one reason or another, have no ability to see it ... Not everyone can ascertain whether 1 + 1 = 2.

1+1 = 2 is not in the slightest bit "self-evident" in the absence of definitions of "1", "+", and "2".

It so happens that 1+1 is the definition of 2. You can't disagree about 1+1=2, without changing the definition of 2 and talking about something entirely different altogether. It would be like insisting the sky is red simply because you refuse to agree on the names of colors.

 

[Yahweh]

You mean like a mobius strip, which has only one side and one edge?

http://www.physlink.com/Education/AskExperts/ae401.cfm