Axioms & Faith: What's the Difference?

[sciencectn]

I had a question about axioms. Assuming I understand this correctly, axioms can neither be proven nor disproven; they are self-evident definitions that we have made to simplify math.

So someone (with a strong religious motivation I might add) said that axioms are based on faith. You can't prove or disprove them, therefore you just have to accept them on faith.

What I'm wondering is, what really separates axioms from other beliefs held on faith? The main differences I can see are that axioms were created in a far more logical way and that they can be changed if necessary (unlike most religious beliefs). But, in terms of supporting evidence, they would almost seem the same as other faith-based beliefs. You can neither prove them nor disprove them.

 

[rootX]

Earth wasn't created 2000 years ago ...

Interesting question and I never thought of that but I think Axioms stand because there is no contradiction to them.

 

[sciencectn]

That's true. I was thinking more about faith based beliefs that can't be disproven (like with Russell's Teapot).

 

[Hurkyl]

In math, an axioms is just a statement, no different from any other statement, aside from the fact we decided to call it an axiom.

A mathematical proof is essentially just an arithmetic calculation, except it's done with statements instead of with numbers.

A mathematical theory is a collection of statements. Typically, we present a theory by naming certain statements as axioms, and then saying the theory consists of all statements that can be proven from the axioms.

 

[Proton Soup]

yeah, if you look at the probability axioms, they're just simple math statements. you can't really say more than that. we can use results to make predictions about processes that appear random to us, but whether the universe is truly deterministic or not is unknowable. but it provides a framework for solving problems.

http://en.wikipedia.org/wiki/Probability_axioms

 

[sciencectn]

So are we essentially taking it on faith that the universe is deterministic?

 

[Jimmy Snyder]

An axiom is in the same form as a theorem, but is not proven, it is taken to be true without proof. Theorems are then proven using the axioms. A set of axioms needs to be consistent and independent. Consistent means that you can't prove both theorem A and theorem ~A. Independent means that if you remove one of the axioms, you can't prove that axiom as a theorem using the remaining axioms.

This does not mean that the axioms cannot be proven, it just means that they aren't. If you pick a different set of axioms, then any axiom in your current set may show up as a theorem.

 

[Gokul43201]

I had a question about axioms. Assuming I understand this correctly, axioms can neither be proven nor disproven; they are self-evident definitions that we have made to simplify math.

So someone (with a strong religious motivation I might add) said that axioms are based on faith. You can't prove or disprove them, therefore you just have to accept them on faith.

What I'm wondering is, what really separates axioms from other beliefs held on faith? The main differences I can see are that axioms were created in a far more logical way and that they can be changed if necessary (unlike most religious beliefs). But, in terms of supporting evidence, they would almost seem the same as other faith-based beliefs. You can neither prove them nor disprove them.

You seem to be confusing axioms in mathematics with postulates/axioms in the physical sciences. Mathematics attempts to make no predictions about the physical world that we interact with. There is no discrimination in Mathematics between one set of axioms and another.

On the question of the difference between postulates accepted by physical scientists and blind faith, there is a huge difference between the two. The former are repeatedly tested through experimental/observational verification of various models that are based upon them. Tests of Special Relativity, for instance, are also tests of the validity of its axioms. We accept certain postulates or axioms, not on faith, but because the models built upon them generate reproducibly good predictions of the behavior of the physical world.

 

[sciencectn]

That clarifies things a bit. But let me give an example of one such axiom believed on "faith", from Euclid's Elements:

"It is possible to draw a straight line from any point to any other point."

He argues that since there is no evidence in nature of a perfectly straight line, we have to have faith in its existence.

 

[Jimmy Snyder]

The theorems follow from the axioms whether you have faith in them or not.

 

[lisab]

The theorems follow from the axioms whether you have faith in them or not.

Is that you, jimmysnyder? I hardly recognized you! You look great!

 

[dx]

That clarifies things a bit. But let me give an example of one such axiom believed on "faith", from Euclid's Elements:

"It is possible to draw a straight line from any point to any other point."

He argues that since there is no evidence in nature of a perfectly straight line, we have to have faith in its existence.

Axioms are implicit definitions. For example, in the mathematical structure that we call 'Euclidean geometry', the axioms are implicit definitions of the words 'line', 'point' etc. So when we say something like "there is a single straight line between any two points", it is not a 'belief based on faith' about the world of experience, but a kind of definition of the terms that occur in this statement. A number of such statements will serve to characterize the whole conceptual structure of Euclidean geometry, and will be a basis for a deductive system. At this stage, there is no question of the 'existance' of any of the schemata that occur in this system, since the mathematics itself does not assert any connection with Nature or experience.

The confusion lies in the relationship of this purely conceptual structure to the world of experience. Now we are outside the domain of mathematics, and enter the realm of physics, which deals with the description of experience. It is of course clear that the only reason these concepts of Euclidean geometry were invented was because they describe our experience, but it is important to realize that this connection between a mathematical structure and experience is not something that can be arrived at a priori. Our immediate experience of navigation on the surface of the planet can be adequately described by the scheme of Euclidean geometry, but a more careful analysis of experience reveals that it is ultimately inadeqaute, and must be replaced by non-Euclidean geometry.

 

[dx]

So are we essentially taking it on faith that the universe is deterministic?

It is not a question of whether the universe is deterministic, but how much of our universe can be described within a deterministic framework. The motions of large inanimate bodies, like spinning tops, planets, baseballs etc. can be described within a deterministic framework. When we go to the level of atoms and elementary particles, we have not been able to find a deterministic framework that describes them, and, in fact, we find that even the concepts of space and time cannot be applied in the usual manner. In a sense, the atomic processes transcend description in the causal/deterministic framework within space and time.

 

[vici10]

Consistency of ZFC (axioms of set theory) cannot be proven within ZFC and therefore it is taken on faith. In other words, we "believe" that axioms will not lead to contradictions.

 

[Hurkyl]

One does not need to believe the axioms of ZFC are consistent in order to use them.

 

[vici10]

If ZFC is not consistent then one can prove any statement.

 

[Hurkyl]

If ZFC is not consistent then one can prove any statement.

That is true^*. But what does it have to do with my assertion?


*: Assuming, of course, that metamathematics is a model of formal logic

 

[vici10]

What is the point to use the axioms if one does not believe in their consistency?

 

[Galteeth]

What is the point to use the axioms if one does not believe in their consistency?

There can be a purpose outside of mathematics.

Evolution has seem to found a use for them. Most human judgements are made from heuristics, which are not consistent, but are consistent enough to be of utility in real world application.

The wikipedia article gives a good summary of the topic.

http://en.wikipedia.org/wiki/Heuristic

 

[zoobyshoe]

Axioms are implicit definitions. For example, in the mathematical structure that we call 'Euclidean geometry', the axioms are implicit definitions of the words 'line', 'point' etc. So when we say something like "there is a single straight line between any two points", it is not a 'belief based on faith' about the world of experience, but a kind of definition of the terms that occur in this statement. A number of such statements will serve to characterize the whole conceptual structure of Euclidean geometry, and will be a basis for a deductive system. At this stage, there is no question of the 'existance' of any of the schemata that occur in this system, since the mathematics itself does not assert any connection with Nature or experience.

The confusion lies in the relationship of this purely conceptual structure to the world of experience. Now we are outside the domain of mathematics, and enter the realm of physics, which deals with the description of experience. It is of course clear that the only reason these concepts of Euclidean geometry were invented was because they describe our experience, but it is important to realize that this connection between a mathematical structure and experience is not something that can be arrived at a priori. Our immediate experience of navigation on the surface of the planet can be adequately described by the scheme of Euclidean geometry, but a more careful analysis of experience reveals that it is ultimately inadeqaute, and must be replaced by non-Euclidean geometry.

Excellent post! I think this clears the issue up perfectly.

 

[Jimmy Snyder]

Axioms are implicit definitions. For example, in the mathematical structure that we call 'Euclidean geometry', the axioms are implicit definitions of the words 'line', 'point' etc.

I don't agree that the axioms are definitions. In geometry, point and line are undefined. The axioms certainly do rule out certain concepts of what point and line might mean. However, they fail to rule in any concepts, for we can never be sure that our concept conforms to all of the axioms. It has never been proved that the axioms of geometry are not vacuous.

 

[Pythagorean]

Axioms aren't self evident. Axioms are something we reserve for our long-term suspended belief.

We accept it to be true for the course of the argument... In the case of mathematics, an argument that has not come to a conclusion.

 

[Gokul43201]

You seem to be confusing axioms in mathematics with postulates/axioms in the physical sciences. Mathematics attempts to make no predictions about the physical world that we interact with. There is no discrimination in Mathematics between one set of axioms and another.

...

That clarifies things a bit. But let me give an example of one such axiom believed on "faith", from Euclid's Elements:

"It is possible to draw a straight line from any point to any other point."

He argues that since there is no evidence in nature of a perfectly straight line, we have to have faith in its existence.

If you do not have a response to this argument, you must not have read my previous post (quoted above), or must not have understood it.

To repeat, mathematics does not attempt to deal with what exists in nature.

 

[dx]

I don't agree that the axioms are definitions. In geometry, point and line are undefined.

How can you prove theorems about points and lines if you have not defined them? As I said, the Euclidean postulates implicitly define what these terms mean. Here, the word 'mean' is not to be interpreted as 'physical meaning', since mathematics by itself does not attach a physical meaning to its elements. 'Line' and 'point' are empty conceptual elements, but mathematically defined nonetheless within Euclidean geometry.

 

[Hurkyl]

What is the point to use the axioms if one does not believe in their consistency?

The same as if one does believe in their consistency.

Maybe there is a misunderstanding? When I say "does not believe in X", that is not the same thing as "believes in not X".

 

[Pinu7]

An axiom is a statement that we assume is true. The collection of axiomatic statements may imply the truth/falsity of other statements. This is called an axiodeductive system. It has nothing to do with religion.

 

[Jimmy Snyder]

How can you prove theorems about points and lines if you have not defined them?

Pick up any high school geometry book. It should start with a discussion of points and lines. It will say explicitly that they are undefined. Then come the axioms. They do not define points and lines either, but they do rule out some commonplace misconceptions such as marks made with a pencil on a piece of paper. The problem with trying to define things is that they must be defined in terms of other things. These other things need to be defined and so on ad infinitum. So to cut the knot, Euclid just gave up and made them abstract.

As to proving theorems about things not defined, you use the axioms (these are theorems not proven). So let's not define foo and bar, and let's accept two axioms:

Axiom 1: There is a foo.
Axiom 2: Every foo is a bar.

Theorem 1: There is a bar.
Can you prove this theorem even though foo and bar are not defined? Note that the axioms do not define foo and bar either, but they do rule out that either of them could be Santa Claus (apologies to our Christian friends). What's more, the axioms might be vacuous. That is, nothing in the real world nor in the world of ideas, satisfies them.

 

[disregardthat]

Axiom 1: There is a foo.
Axiom 2: Every foo is a bar.

Theorem 1: There is a bar.

Both axioms are nonsense to me. I don't believe this is a proper analogy to axioms in mathematics.

 

[Jimmy Snyder]

Both axioms are nonsense to me. I don't believe this is a proper analogy to axioms in mathematics.

There is no requirement in mathematics for the axioms to make sense to you, that is your resposibility alone. Your beliefs do not enter into it either. All that is required is that they be consistent and independent.

 

[Pinu7]

Both axioms are nonsense to me. I don't believe this is a proper analogy to axioms in mathematics.

In which way? They seem fine to me.

 

[disregardthat]

In which way? They seem fine to me.

The axioms as premises are fine in themselves, but it is their mathematical equivalence I protest.

Axiom 1 is analogical to an axiom which states the mathematical existence of a mathematical object in a non-constructive way. That is absurd and contradictory to me. I don't buy into the game in which mathematics is the meaningless play with symbols, and mathematical models is detached from this. For me, mathematics is what we call mathematical models where statements makes sense.

 

[dx]

Pick up any high school geometry book. It should start with a discussion of points and lines. It will say explicitly that they are undefined. Then come the axioms. They do not define points and lines either, but they do rule out some commonplace misconceptions such as marks made with a pencil on a piece of paper. The problem with trying to define things is that they must be defined in terms of other things. These other things need to be defined and so on ad infinitum. So to cut the knot, Euclid just gave up and made them abstract.

As to proving theorems about things not defined, you use the axioms (these are theorems not proven). So let's not define foo and bar, and let's accept two axioms:

Axiom 1: There is a foo.
Axiom 2: Every foo is a bar.

Theorem 1: There is a bar.
Can you prove this theorem even though foo and bar are not defined? Note that the axioms do not define foo and bar either, but they do rule out that either of them could be Santa Claus (apologies to our Christian friends). What's more, the axioms might be vacuous. That is, nothing in the real world nor in the world of ideas, satisfies them.

If you take your two axioms to be the definition of a mathematical structure, then yes, those axioms do define 'foo' and 'bar', in the sense that any property of these elements, as they occur in this structure, is implicitly contained in the axioms.

A given mathematical structure, such as Euclidean geometry, can be axiomatized in many ways, but for a set of propositions to be considered axioms, they must completely define the structure. All properties of the concept of 'line' are implicitly contained in the axioms, and therefore the axioms implicitly define the idea of 'line', and also implicitly define the whole scheme of Euclidean geometry. Here, by 'define', I mean mathematically defined.

The reason I brought this up in the first place was to show that the Euclidean axioms about lines and points are not to be thought of as 'self-evident truths', or 'things accepted on faith'. The axioms are simply to be thought of as the definition of these concepts (implicit definitions, to be precise). This removes the mystery surrounding the foundations of geometry, where people originally thought that the axioms are somehow a prori "truths", and the reason for this being that they could not distinguish between the mathematical/conceptual scheme of Euclidean geometry, which by itself is physically vacuous, and their intuitive and tacit conversion of this into a physical theory connected with their experience.

 

[Hurkyl]

At this point, I think you're debating over your favorite meaning of an English word, rather than over mathematics.

 

[Hurkyl]

All that is required is that they be consistent and independent.

Actually, neither of those is required of axioms either: first is merely a desirable property, and it is often useful to violate the second.