When I referred to axioms previously (including in that post you quoted), I was referring to axioms in the context of philosophy (metaphysics/epistemology as opposed to mathematics), which was the context in which the original quote occurred. The wikipedia article seems to focus on the mathematical/physics application (ie. as a starting assumption) of the word.

Context is key here.

Fixed

Again, different context. Given that most encounter axioms in Secondary math classes, it is understandable.

Please see previous posts re: definition of axiom.

Ummm, what "religious discussions" are you referring to? My response to a response to yet another response? The mere reference to religious figures does not a "religious discussion" make.

@utabintarbo: Just said that about religion as a precaution. In your quote of a definition of an axiom:
I disagree, you can take axioms that need other axioms to define things, but cannot be proven from that. Also note that some axioms can either be taken as axiom, or as proof from other (stronger)axioms; and there is also equivilence. I would instead say "Axioms are not proven from anything to be valid", although sometimes internal consistentcy is not obvious.
An aspect of logic; A=> not A is a proof that A is false by self-contradiction. An axiom that self-contradicts isnt much use .
BTW Asides: I hardly encountered all my axioms in secondary classes; My profile reads: bothed physics student. In fact, i did some extra math on the side, and my first 2.5years i were a perfectly good student. (I am taking a in-between year to maybe be one again.)

I don't believe in this sort of thing. I'm a skeptic, and I don't believe in any kind of "certain" knowledge.

I also don't believe in such a thing as an axiom that I "can't deny without using it in my denial."

Unless by axiom you mean a paradox, like, "this sentence is false"...to deny that I have to affirm it.

I think that kind of thing is cool, but I don't see how to build knowledge on it.

God only knows what I saw in secondary math classes.

However, I *like* the wikipedia definition of axiom better. As I said, I don't believe there is such a thing as a "certain" or "irrefutable" starting point. I think starting points can't be proven, only assumed...in that case, they can be arbitrary.

I'm looking for plausible ones (plausible to me), not arbitrary ones. Or maybe I'm looking for aesthetically pleasing ones? At any rate, I'm not looking for axioms which satisfy some inarguability criterion, because I don't think it's possible.

"Axioms" derived from other axioms fail the test of axioms in that they would no longer be irreducible primaries. This topic is about axioms.

Precisely my point.

I'm sorry to hear that, but you probably don't believe that either.

I don't believe that you really don't believe that. But then you likely don't believe that I don't believe that you don't believe it.

Rampant skepticism gives me a headache.

No.

To a skeptic who does not believe in certain knowledge, all knowledge must necessarily be arbitrary. Thus is the self-refuting nature of skepticism.

You used the word "must" so he doesn't believe you.

Seriously, I must commend you on a very skillfully-done straw man.

Some (like Heraclitus) would say "we can't step in the same river twice," as it's constantly changing. IOW, the identity of the river is never the same at any two points in time ... and that any grouping of river-states under a single model is just an abstraction that isn't *really* real. The thing that's real is the constant changing components, the water molecules.

The same could be said of one's own mind... as the physical states and thoughts are constantly changing. The unified illusion of persistence over time is an abstraction of our own observations of ourselves. In a sense persistent identity implies some sort of global unity and static nature.

There's an element of this thought in Buddhism. Buddhism really cuts to the heart of many assumptions about language, and truth (even rejecting two-valued logic in favor of fuzzy logic).

But what's radically different in Buddhist thought, though, is there's not as much emphasis on believing one single thing. A Buddhist might say there are two correct ways to look at the picture, in that in one sense a river is a real entity, and in another sense it is not. That both ways of seeing it form the complete picture.

Personally, I prefer a more round-about way of speaking about an entity like an abstract conceptual model ... perhaps some elements of Buddhist thought have worn off on me. A (Platonic?) statement like "everything that exists has a specific nature" seems like it's saying that the thing exists first before the properties. But IMO, it's only because we group similar perceptions under a common model ... like a river.

@yaaarrrgg: That's pretty interesting. I would still say the river exists. It is dynamic, and the idea of river is abstract, but it's still a river. So it still exists and has some kind of properties/substance to it.

I would definitely agree that there is no one way to see things. That is, I believe that absolute truth exists, but it's too large for any human to be able to grasp it all. There are infinitely (uncountably even) many truthful ways to see something. Each instant of the river is truly the river, but not the whole river. It would take every moment of the river's existence to compile the entire truth of the river.

Thanks for posting.

Nicely posited.

I'd like to comment on this, but I need to think about it first.

I don't, but not because of skepticism. I disbelieve it because of your propensity towards sarcasm.

I believe that you don't believe that I really don't believe...OK, I'm out.

I believe lots of things. Mainly, I believe the obvious things in front of my face. If you say you are exactly 2 meters tall, I believe you. I'm even a bit naive about it.

What I *don't* believe in is absolute certainty. I'm open to the possibility that I might be wrong.

In particular, I don't think there's such a thing as a self-evident, or self-justifying, or Knowledge-Giver-Justified Axiom.

I take the ones that make the most sense to me...but how--honestly, HOW--how can ANYONE with any shred of intellectual honesty claim that their axioms are unequivocally, irrefutably, universally true, beyond any possibility of doubt?

I mean, I may *think* mine are true, but how the heck could I know for sure?

Bnonn has scared me out of using the word 'knowledge,' but my beliefs are not arbitrary. My axioms are not arbitrary. As I've said, I try to stick to plausibility, simplicity, and beauty in my axioms.

I just think that axioms are *allowed* to be arbitrary, by definition. Doesn't mean the ones I choose to believe about the universe are *arrived at* arbitrarily.

And you prefer the self-affirming nature of dogma?

But that is not what i meant, axioms can rely on other axioms without it being possible to prove the newly defined axiom from the older one. For instance, uniformity of nature first assumes nature exists. (if you want to call those postulates, very well, we'll talk postulates then.)
It may be possible to turn sets of axioms depending(not provable from) on eachother into sets that do not. But it may be difficult to do so, and hinder discussion. I do not see axioms depending on(/but not provable from) eachother as a problem.
About "A=> not A is a proof that A is false", i have not yet seen that problem here yet. What i meant by saying that, is that there was unnessesary redundancy, although now i see the definition was only the above six lines of quoted link. I think that we should use wikipedias axiom instead.

hope we can back to axioms themselves

Right here is the crux of the matter...while the specific instance of that river may never be exactly the same again, the concept of "river" is (relatively) unchanged. It exists as an epistemological abstraction within the consciousness of those that know of that river (or of rivers in general).

Who? Me?

Are you absolutely certain that absolute certainty is impossible? I, too, am open to the possibility that you might be wrong.

Try to refute "existence exists" without conceding that existence does exist. Try to refute the Law of Identity while you're at it.

By what standard?

Please review the definition of "axiom" above.

Obviously, you can't. Do not, therefore, attempt to hold the rest of humanity down to your level.

That anyone would allow that irrational troll to scare them out of anything gives me doubts as to the intellectual fortitude of those in fear.

It is not enough to cast doubt on all knowledge, you find it necessary to attack metaphysics as well? Have you no boundaries?


You find my intellectual certainty so threatening that you label it as "dogma"?

I really do not know about this. Discussing this requires some kind of judgement about which axioms to discuss. Simplicity of axioms and, for sets of axioms, unability to derive them from eachother, seems to be the only measure of that. We can only hope that is enough to keep some focus on a few axioms.
Things that cannot be proven wrong need not be true.(have you put enough thought in that part of your post?) For some mathematical things it is actually proven that they cannot be proven false and also not cannot be proven true. These are axioms, precisely because they cannot be proven. (sometimes axioms can be proven by taking other axioms though, said something in this vain earlier.)
I agree, but it is not like people who think absolute certainty is true are ever going to convince me .
Eh, your confidence is great, but please keep an open mind about the possibility you are wrong. The discussion is not a contest either, so, at some point of the disagreement everything has been said; Then agree to disagree, so that some more consideration, may clarify.
The every-thing-consistent-behavior does not specify if these things are somehow part of each other, does not have a problem with it. I earlier considered the external world just one of these things, this seemed nessesary to be able to have properties like "uniformity of nature".
The "Buddhist model" seems to take the opposite view in that the assumption is that nature(external world) exists and all things in it are just abstractions we make. I think these abstractions can be handled similarly as the axiomatic every-thing-consistent-behavior view. (BTW repeating others a little here)

It seems to me that we can claim the existence of things, without restricting the nature of that existence. A river exists. That can be true whether we consider the river to be a physical object, or merely an abstract concept, or an amalgamation of an infinitude of physical objects. It still exists. Therefore I propose a statement of an existence axiom which allows for multiple interpretations of the physical world, all without actually denying existence.

Something like this:
- The world around us does, in fact, exist external from our minds.

- Everything in the world has a specific nature. That nature need not be static, nor observable, yet it does exist.

I'm not thoroughly satisfied with that, but it's what I came up with on the spot. Please feel free to adjust it.

Well, let me take a shot at it

(1) existence exists

this unpacks to

(1') that which exists, exists

This however is a bit circular, don't you think? It doesn't really shed any light on whether or not anything really exists.

What you want to say is:

(1'') At least one thing exists, and it's not "me"

But this is different than 1, and can't be determined from a purely logical point of view.

As for the law of identity:

(2) A is A

This is a tautology. From it you can only derive B is B, C is C, D is D, etc. You cannot derive the law of non-contradition, or even that A is not B. This in itself is not very useful.

Also, this is different that the metaphysical claim:

(2') Everything that exists has a specific nature.

This can be taken as an assumption or not... there's nothing *logically* binding here.

Axioms can't really pull themselves up from the bootstraps. They are just glorified assumptions.

"The" definiton, or Ann Rand's definition? Generally, I just use the term to mean any starting assumption (or rule).

Perhaps, but that is precisely why I have limited myself to three (3) axioms. These can neither be refuted nor denied (or else all this is moot).

And please take heed of the differences between the Mathematical and Philosophical defintions of "axiom".

Ad they call me dogmatic....

Life itself is a contest. If it should come to that, I will agree that you are wrong, and dismiss you out of hand. </arrogance>

For example, the Axiom of Choice.