Are axioms beliefs?

[Dorian Gray]

I say that in the sense that everything is a belief, axioms are a belief. Not a religion, though.

Now, if you were to ask if axioms were religions, man would this thread get long.

I apologize in advance, although this apology will eventually be forgotten after a few dozen pages.

 

[Yahweh]

This is the first time I've noticed, but Dorian Gray's avatar is actually a Mobius Strip also...

It is a crazy world we live in...

 

[Dorian Gray]

This is the first time I've noticed, but Dorian Gray's avatar is actually a Mobius Strip also...

It is a crazy world we live in...

But mine is smooth and round and shiny, not all pointy and daggery like yours.

 

[DrMatt]

Axioms are sentences that are used as premises for arguments. Using an axiom does not require that you agree that the axiom is true.


AXIOM. The angles of a triangle add up to pi.
This is one of a number of axioms which lead to:
THEOREM: For a right triangle, the square of the length of the hypotenuse is equal to the sums of the squares of the lengths of the bases.

I can compute with this axiom without believing one way or another about it. I can also compute with non-Euclidian geometries in which this axiom is not present.

AXIOM: Harry Potter is immortal.
AXIOM: All Men are mortal.
AXIOM: Anything that is immortal is not mortal.
THEOREM: Harry Potter is not a man.

I can compute this theorem without having any belief about the axioms at all. In fact, since there is no evidence that the Harry Potter referred to in the first axiom actually exists anywhere, the axiom itself is incoherent and not the sort of thing about which I can have any belief for or against.

Axioms are not beliefs.
Person P believes Sentence S when P honestly declares that S is true. Merely supposing S as an axiom in order to derive its logical consquences is a different thing.

 

[BillHoyt]

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.

That would be "strip club bouncer." Here is what you said, Tr'olldini:

"Axioms = beliefs = assumptions"

Of course, you left out the full quote to disguise its essential equivocation. This is precisely the same fallacious reasoning as used in "science = faith = religion." Assumptions can be founded or unfounded. Beliefs can be faith-based or fact-based. The axioms of science and mathematics are well-founded. They remain as axioms not out of faith, not out of lack of evidence for them, but out of lack of underlying principles from which to derive them or direct tests to falsify them. They remain totally supported by the theorems derived from them (in the case of mathematics) and by the theories and findings of science. They are simply not derivable from first principles (mathematics). In the case of science, the axioms are indirectly tested with each experiment. There is no direct test for them. That is why they remain axiomatic.

 

[DrMatt]

I'd just like to mention that a math degree is quite besides the point on this topic, as the distinction is one of rudimentary philosophy of language and not particularly a mathematical object.

Matt Fields, BA Math, BMus composition, MA music, DMA composition

 

[T'ai Chi]

The axioms of science and mathematics are well-founded.

We might have reason to suppose they are true (that's what we do with axioms), but they are still beliefs.

 

[BillHoyt]

We might have reason to suppose they are true (that's what we do with axioms), but they are still beliefs. [/B]

Equivocation. Again. PoMo equivocation nonsense. Still fallacious, and doesn't improve with repetition.

 

[T'ai Chi]

Equivocation. Again. PoMo equivocation nonsense. Still fallacious, and doesn't improve with repetition.

I have no idea what "PoMo" has to do with this. I do know that you haven't demonstrated evidence for your belief of why axioms are supposedly not beliefs. Again.

You may do so in your next post.

 

[Clancie]

T'ai,

An axiom may, in reality, be a belief, but axioms can't be defined as beliefs.

I think the confusion comes because the word "axiom" is used differently in mathematics and in philosophy.

In philosophy, an axiom is a statement that is accepted as obviously being true without anyone actually needing to prove it's true. You accept these axioms as true and then can use them to make statements about what you know then about other things.

In math, an axiom is just a given premise--it may or may not be true. But, whether the axiom is true or not true, that still doesn't let anyone claim that "an axiom" means just the same thing as "a belief". That's changing the definition to indicate something that isn't there.

Here's an example of the relationship between truth (as opposed to belief) and axioms, from a Stanford website:

An analytic definition associates with the constant being defined a defining axiom. Intuitively, the meaning of a definition is that its defining axiom is true and that its defining axiom is an analytic truth. Analytic truths are considered to be those sentences that are logically entailed from defining axioms. For example....

Stanford.edu

 

[Clancie]

Posted by T'ai Chi

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

If you substitute the word "truth" for the underlined word, you'll see the problem.

(And I don't think you'd want to argue that all axioms are beliefs, unless you want to argue that everything we think we know is a belief).

 

[BillHoyt]

T'ai,

An axiom may, in reality, be a belief, but axioms can't be defined as beliefs.

I think the confusion comes because the word "axiom" is used differently in mathematics and in philosophy.

In philosophy, an axiom is a statement that is accepted as obviously being true without anyone actually needing to prove it's true. You accept these axioms as true and then can use them to make statements about what you know then about other things.

In math, an axiom is just a given premise--it may or may not be true. But, whether the axiom is true or not true, that still doesn't let anyone claim that "an axiom" means just the same thing as "a belief". That's changing the definition to indicate something that isn't there.

Here's an example of the relationship between truth (as opposed to belief) and axioms, from a Stanford website:

Stanford.edu

Red letter day. Caloo, calay!

 

[T'ai Chi]

Axioms are sentences that are used as premises for arguments. Using an axiom does not require that you agree that the axiom is true.

AXIOM. The angles of a triangle add up to pi.
This is one of a number of axioms which lead to:
THEOREM: For a right triangle, the square of the length of the hypotenuse is equal to the sums of the squares of the lengths of the bases.

I can compute with this axiom without believing one way or another about it. I can also compute with non-Euclidian geometries in which this axiom is not present.

AXIOM: Harry Potter is immortal.
AXIOM: All Men are mortal.
AXIOM: Anything that is immortal is not mortal.
THEOREM: Harry Potter is not a man.

I can compute this theorem without having any belief about the axioms at all. In fact, since there is no evidence that the Harry Potter referred to in the first axiom actually exists anywhere, the axiom itself is incoherent and not the sort of thing about which I can have any belief for or against.

Axioms are not beliefs.
Person P believes Sentence S when P honestly declares that S is true. Merely supposing S as an axiom in order to derive its logical consquences is a different thing.

Oops, I forgot to state that I am specifically talking about axioms in mathematics.

 

[T'ai Chi]

Red letter day. Caloo, calay!

LOL.

Anyway, from http://www.hyperdictionary.com/dictionary/Axiom we get that an axiom is:

"a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident "

"A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted"

"A well-formed formula which is taken to be true without proof in the construction of a theory."

Clancie, the notion of unproved truth makes no sense. If it is unproved, how do you really know it is a truth? As Billy said, you might test it and have it confirmed a lot, but that is not the same thing as proving something true.

 

[Clancie]

Posted by T'ai Chi

Oops, I forgot to state that I am specifically talking about axioms in mathematics.

Okay, but then you say this, (which is "axiom as epistemology", not "axiom as math")...

Posted by T'ai Chi

"A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted"

That's philosophy, not math.

Posted by T'ai Chi

Clancie, the notion of unproved truth makes no sense. If it is unproved, how do you really know it is a truth?

Okay, you're back to epistemology again. I think I understand what you're saying re: the problem you see in building knowledge philosophically based on axioms that are unproven. Is it kind of, "How can you say you're building knowledge based on premises that may seem self-evident yet, in reality, only appear to be true?"

If that's it, all I can say is axioms, by definition, are premises we philosophically accept as true in order to construct further knowledge.

Now you may say, "Isn't a 'premise' that appears true but may not be true, just the same thing as a 'belief'?"

You could probably get some people to agree with you, but to me, as I said before, an axiom may be a belief (or it may not), but if you reason from axioms about the nature of the world, you are accepting the definition of them--and the definition doesn't allow you to assume that an axiom = a belief. (Maybe the problem is also somewhat linguistic--all the meanings loaded into the word "belief"....)

 

[BillHoyt]

Clancie, the notion of unproved truth makes no sense. If it is unproved, how do you really know it is a truth? As Billy said, you might test it and have it confirmed a lot, but that is not the same thing as proving something true.

That "notion" lays at the foot of all logic. Logic takes it as axiomatic that a thing that is A is not also Not-A. Without that axiom, we can't rub two premisses together to start a conclusion.

 

[T'ai Chi]

I posted the following dictionary definition:

"A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted"

Clancie posted:

"That's philosophy not math, T'ai."

No, that is from math. In fact, if you actually read the page I provided or read it again, it, in its entirety says:

"1. (Logic & Math.) A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted; as, ``The whole is greater than a part;'' ``A thing can not, at the same time, be and not be.''

I left out the "(Logic & Math)" and the trailing part to specifically see how many people confuse one definition of axiom with another. .. and sorry, you confused the logic and math definition of axiom with the philosophy definition of axiom.

 

[T'ai Chi]

That "notion" lays at the foot of all logic. Logic takes it as axiomatic that a thing that is A is not also Not-A. Without that axiom, we can't rub two premisses together to start a conclusion.

That's fine, sir. But that doesn't mean that axioms are not beliefs.

 

[BillHoyt]

That's fine, sir. But that doesn't mean that axioms are not beliefs.

I've been watching this No True Scotsman maneuver with glee for a while now. That is not what you claimed at the outset. You claimed an equivalence. Here you are shifting into a claim that they are subsumed under "beliefs."

 

[Clancie]

Posted by T'ai Chi

1. (Logic & Math.)

T'ai,

I'm not trying to argue with you (although I disagree completely ), just giving you my response to your question.

However...imo...the fact that your dictionary felt it was okay to lump "Logic and Math" together with a single definition for "axiom" without differentiating between them, just kind of highlights how inadequate dictionaries often are when it comes to more subtle philosophical discussions.

"...illustrates A self-evident and necessary truth...''

I don't care what that dictionary says, this is epistemology. This isn't the correct use of "axiom" in math. An axiom in logic, in math, in philosophy are just not the same.

That said, I haven't read anything that contradicts what I posted to you above about the use of "axiom" or why you can't make it synonymous with belief.

Guess we'll just have to "agree to disagree!"