What is the justification for "axioms"?

[tdgesq]

The axiom word has been invoked multiple times (particularly by non-Catholics) on this forum. I understand the definition of the word “axiom.” What I want to know is the justification required to declare a proposition an “axiom.”

 

[Oreoracle]

The axiom word has been invoked multiple times (particularly by non-Catholics) on this forum. I understand the definition of the word “axiom.” What I want to know is the justification required to declare a proposition an “axiom.”

In every system, there are principles we accept without question. In math, for example, we assume that A=A, that if A=B then B=A, and so on. These principles must be assumed for mathematics to exist as a system. Or you could think about it in terms of words. Try this: First, look up a common word in the dictionary. Usually, such words are defined through the use of synonyms (which isn’t very effective). If the word in question is defined by synonyms, look up one of the synonyms. Repeat this process, and you will eventually find the original word that you looked up before being used to define one of the consequent words. Definitions are often circular, and so we treat some words/phrases as though they’re self-explanatory, for the sake of the language, just as we assume mathematical properties, postulates, and theorems for the sake of mathematics.

In short, axioms are concepts used to construct logical systems. Thus, any attempt to prove them by using their own system will result in circularity.

 

[James_S_Saint]

The axiom word has been invoked multiple times (particularly by non-Catholics) on this forum. I understand the definition of the word “axiom.” What I want to know is the justification required to declare a proposition an “axiom.”

You, nor anyone can think without them.

If effect, an axiom is “the beginning”. How can you construct a thought without beginning?

 

[warpspeedpetey]

The axiom word has been invoked multiple times (particularly by non-Catholics) on this forum. I understand the definition of the word “axiom.” What I want to know is the justification required to declare a proposition an “axiom.”

in the simplest form, “necessity.”

we need a starting place to begin a logical exploration of the world.

 

[Eucharisted]

An axiom is a self-evident truth.

 

[tonyrey]

The word “axiom” comes from the Greek word ἀξίωμα (axioma), a verbal noun from the verb ἀξιόειν (axioein), meaning “to deem worthy”, but also “to require”, which in turn comes from ἄξιος (axios), meaning “being in balance”, and hence “having (the same) value (as)”, “worthy”, “proper”. - wikipedia

We could infer from this that an axiom is a worthy, valuable, proper and balanced requirement for reasoning…

 

[Luke_K]

So since you cannot prove an axiom, does that mean it’s held on faith? And in that case by extension, wouldn’t all of our knowledge be held from faith?

 

[tonyrey]

So since you cannot prove an axiom, does that mean it’s held on faith? And in that case by extension, wouldn’t all of our knowledge be held from faith?

Initially it is on faith - since there is no way of proving it - but if it turns out to be constantly successful it is granted the status of a “necessary truth”. At least that is my interpretation until a better solution is suggested…

 

[geometer]

Initially it is on faith - since there is no way of proving it - but if it turns out to be constantly successful it is granted the status of a “necessary truth”. At least that is my interpretation until a better solution is suggested…

So the use of axioms is completely dependent of our knowledge of experience. Axioms are shown to be true through our successful applications.

Where have all the axioms come from?

 

[hatsoff]

An axiom is a self-evident truth.

A lot of people, including some (but certainly not all) mathematicians, will say exactly that, but I consider it a poor and ultimately false description. There is nothing self-evident about many axioms.

Axioms are true because we declare them to be true. Thus it becomes important to have some good reason for imposing such a declaration. For example, if it is useful to treat a certain proposition as true, and if there is no way to prove that it is true, then we might simply declare it to be true as an axiom.

 

[hatsoff]

So since you cannot prove an axiom, does that mean it’s held on faith? And in that case by extension, wouldn’t all of our knowledge be held from faith?

Axioms are used in logical systems. If we treat knowledge as one big system, then that’s called foundationalism, in which case we have a special term, basic beliefs, which describe the core beliefs on which all other beliefs are ultimately based. A basic belief is a lot like an axiom, but it applies to epistemic systems instead of logical systems.

If you’re a foundationalist, then you might characterize basic beliefs as “faith”-based. However, such a description might not be very accurate, since the word “faith” has religious overtones inappropriate for epistemology.

Furthermore, we must ask the question, is foundationalism accurate? In my opinion, as well as that of many others, it is not accurate at all.

 

[tonyrey]

So the use of axioms is completely dependent of our knowledge of experience. Axioms are shown to be true through our successful applications.
Where have all the axioms come from?

Philosophers like Plato and Aristotle who examined the nature of knowledge and reasoning. The first systematic treatise on logic, the Organon, was compiled by Aristotle’s students.

 

[tonyrey]

Furthermore, we must ask the question, is foundationalism accurate? In my opinion, as well as that of many others, it is not accurate at all.

What do you replace basic beliefs with?

 

[geometer]

Philosophers like Plato and Aristotle who examined the nature of knowledge and reasoning. The first systematic treatise on logic, the Organon, was compiled by Aristotle’s students.

Where did Aristotle say that his axioms came from? Was he pious at all?

 

[geometer]

What I want to know is the justification required to declare a proposition an “axiom.”

In geometry an axiom is a useful geometric idea which can not be derived from other axioms. If a proposition can be derived from known axioms I think it is called a postulant, and if a deductive proof can be found for the postulant, then it becomes a theorem.

Theorems are quite useful…

 

[tdgesq]

Thank you for all of the responses. I am entertaining guests this week, so I have a limited amount of time to respond. I will try to reply to as many responses as I can.

In short, axioms are concepts used to construct logical systems. Thus, any attempt to prove them by using their own system will result in circularity.

True, but all this recognizes is that any attempt to justify a system within that system will be circular. In informal logic, begging the question in this way is actually a fallacy. The question is: what is it that justifies an axiom?

You, nor anyone can think without them.

If effect, an axiom is “the beginning”. How can you construct a thought without beginning?

Again, this is true, but how does it provide a justification for axioms? First, not all axioms are necessary to construct a rational thought. There are many mathematical axioms that fit this description. Second, is it really a justification to say that without certain axioms (such as those that form the basis of the laws of logic) we couldn’t construct a thought? Perhaps our thoughts are just nonsense, even though we believe otherwise. Perhaps all of this typing back and forth on the philosophy forum is more of the same. This is the skeptic’s challenge.

in the simplest form, “necessity.”

we need a starting place to begin a logical exploration of the world.

What makes that necessary though? If I accept all of the laws of logic except modus ponens, surely there are some things I can know. If I accept all laws of thought but reject induction in the vein of Hume, there are still things I can know. What type of “necessity” suffices for a proposition to qualify as an axiom?

An axiom is a self-evident truth.

I know the dictionary definition of “axiom.” What I want to know is what makes those propositions “self-evident.” I want the justification that demonstrates it is self-evident. Not even the laws of logic are self-justifying, as Oreo alluded to above.

So since you cannot prove an axiom, does that mean it’s held on faith? And in that case by extension, wouldn’t all of our knowledge be held from faith?

That is a very good question. It seems to be that way. If logical laws, induction, the reliability of the senses, etc. are not justifiable, then it appears they are accepted on faith.

Initially it is on faith - since there is no way of proving it - but if it turns out to be constantly successful it is granted the status of a “necessary truth”. At least that is my interpretation until a better solution is suggested…

Maybe. I think the more interesting question is what does “constantly successful” mean. And even more to the point: when do we decide that a proposition has been successful enough to qualify as an axiom?

So the use of axioms is completely dependent of our knowledge of experience. Axioms are shown to be true through our successful applications.

You mean they have been “successful.” There is nothing that ensures or even indicates their future success. That is the problem of induction – assuming that our senses are reliable in the first place.

A lot of people, including some (but certainly not all) mathematicians, will say exactly that, but I consider it a poor and ultimately false description. There is nothing self-evident about many axioms.

I agree.

Furthermore, we must ask the question, is foundationalism accurate? In my opinion, as well as that of many others, it is not accurate at all.

Well, there is always the coherence theory of knowledge if you don’t like foundationalism. I didn’t find either of them very satisfying explanations.

In geometry an axiom is a useful geometric idea which can not be derived from other axioms. If a proposition can be derived from known axioms I think it is called a postulant, and if a deductive proof can be found for the postulant, then it becomes a theorem.

Theorems are quite useful…

You mean theorems have been quite useful in the past assuming that our senses are reliable.

I don’t want to get too sidetracked here. What I am really looking for is when and for what reason a proposition gets the status of an “axiom.”

 

[warpspeedpetey]

the necessity of common ground i suppose, i dont think there are hard and fast rules for the the transition from proposition to axiom, i may be wrong though.

 

[geometer]

You mean theorems have been quite useful in the past assuming that our senses are reliable.

I don’t want to get too sidetracked here. What I am really looking for is when and for what reason a proposition gets the status of an “axiom.”

Well, in euclidean geometry it is taught that straight lines are infinite in extention, this is an axiom. This is never proven, it is assumed to be true. This is also beyond realization for our visual senses and imagination.

This is part of the establishment of the intuition, its one of the highest ranking rules of the game. There are other kinds of geometry which have other axioms concerning lines, like spherical geometry, where every line is a great circle of a spherical surface.

A different intuitive rule a different imagination game…

 

[Oreoracle]

True, but all this recognizes is that any attempt to justify a system within that system will be circular. In informal logic, begging the question in this way is actually a fallacy. The question is: what is it that justifies an axiom?

Oh, I think I see what you mean now.

To put it simply: nothing comes before axioms, and so nothing can justify them in the sense that you (might) mean. However, the systems of mathematics and the scientific method hold high utility values. The fact that these systems are so successful demonstrates that the axioms within them must be worthwhile, or that they describe reality in an adequate fashion, enough so that we can use them to progress with, say, technology. So in this sense, the axioms are justified by their high utility value or success rate. I suppose you could counter by asking: what tells us that the axioms are worthwhile before they are used? For now, that stumps me.

 

[tdgesq]

Not to ignore intervening posts, but let me propose two axioms:

1. God exists as the ultimate cause of being; and
2. morals are objective.

Any comments or objections?