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I am an engineer. Lately I've been thinking a lot about mathematics and how it relates to reality. So naturally I originally posted this question on the mathematics forum. But I was basically run out of town. So I'll try here.

According to Bjarne Stroustrup, "an axiom is something we can’t prove. It is something we assume to be true."

According to Merriam-Webster one of the definitions of faith is, "firm belief in something for which there is no proof".

The definition of assumption is, "a thing that is accepted as true or as certain to happen, without proof."

When I use the term "faith" I use it in the secular sense. For example, in the morning when I see a mug of coffee before me, I have faith that what my eyes are perceiving, and what my brain is interpreting, is in fact a coffee mug.

So if the foundations of mathematics are based on faith, don't mathematical proofs also have to have the same faith in these same axioms? Therefore, do you need faith in order to prove something?

Are faith and proof somehow interrelated?

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    I think that your use of faith is a little misguided and could be replaced by "assumption" because that's more of what is going on. Sure you can say that you need faith to make an assumption but (as you pointed out) the word faith has connotative baggage associated with it that is unnecessary to the conversation. Yes, axioms in mathematics are assumed without proof, but the entire history of mathematics has had a constant stream of arguments for why the axioms we assume are self evident. There is no assumption in a proof, things follow deductively. But the initial axioms are assumed. – Not_Here Jul 3 '17 at 2:01
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    It is complicated without getting into specifics, but the process of figuring out which axioms to assume without proof is part of the foundations of mathematics and we have been doing that for as long as we have been doing math. Euclid chose his postulates because they are self evident and common sensical, so he assumed them without proof. That doesn't mean that those statements are unprovable in general, you could have another system of different axioms that prove his postulates as theorems, but we choose his to use because of their sense of being self evident, almost definitional. – Not_Here Jul 3 '17 at 2:03
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    Math,like any kind of "argument" (or discourse), requires starting points; "an axiom is something we assume to be true", i.e. a "starting point" for a math theory or discipline. We cannot prove it ? not necessarily: Peano's axioms for arithmetic are theorems (i.e.proved) in set theory, of course assuming new axioms (those of set theory). – Mauro ALLEGRANZA Jul 3 '17 at 5:55
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    In the historical development of math, many "obvious" math facts previously assumed as true (e.g. a+b=b+a) has been proved, i.e. it has been showed that they are theorems of some theory standing on some fundamental assumptions, the axioms of the theory. – Mauro ALLEGRANZA Jul 3 '17 at 5:57
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    In conclusion: the assertion "axiom cannot be proved" is incorrect, if asserted without qualification: axioms canot be proved in the context of the theory of which they are the axioms. – Mauro ALLEGRANZA Jul 3 '17 at 5:58
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The Terminology of Faith vs. Assumption

First, I will try to explain why the word assumption is better than the word faith in this context. In the definition of faith you have, it ends with "for which there is no proof" and this, ultimately, is the issue. Faith, being defined in that way, means that it applies to things that cannot, under any circumstances, have a proof. However, the axioms we use in mathematics do not fall under that category. Axioms are relatively unprovable. This means that a statement, such as "parallel lines never intersect on a flat surface," might be assumed without proof in one context, but is completely provable in another context. I'll explain this with an example below, but the point to take away is that there are proofs of mathematical statements that we assume as axioms, but those proofs are in systems that do not use those statements as axioms.

The reason why the statement "we take it on faith that our axioms are true" is less true than "we assume our axioms are true" is because faith says "we are saying there is no proof that parallel lines on a flat plane never intersect but we believe it anyway" when assumption says "we are assuming that parallel lines on a flat plane never intersect without proving it". We don't take it on faith that Euclid's postulates are true, we assume them without proof because they are arguably self evident, or even definitional. The difference comes down to the difference between "there is no proof" and "without proof".

An Example of the Relativity of Proof

As I said above, a certain mathematical statement may be assumed as an axiom in one context (in one specific theory) or it may be derivable as a theorem in another context. Let's take for example Euclid's first postulate:

Any straight line segment can be drawn to adjoin any two points (on a flat surface).

If you are trying to develop Euclidian geometry, this is one of the assumptions you need to make. This, along with his other postulates, are assumed without prior proof and are used as the starting points for logical deductions to be made. We start off assuming that a collection of things are true about lines and points and we then use deductive proofs to prove theorems. Some examples of theorems that Euclidean geometry proves (things like the Pythagorean theorem) are found here.

In current foundations of mathematics we use the axiomatic system ZFC set theory. ZFC is also a list of axioms equipped with a proof system that lets us derive theorems, however ZFC is much more powerful than Euclid's postulates. ZFC tells us that sets, a certain amount and kind of them, exist and that those sets have certain properties. The method of using set theory as a foundation for mathematics is done by treating the sets that ZFC talks about as numbers and that the properties that the sets have reflect the properties of actual numbers. Doing this, ZFC is able to prove (almost) all mathematical statements about geometry, algebra, number theory, analysis, and so on. What ZFC cannot prove is its own axioms. However, ZFC can easily prove Euclid's first postulate once a suitable definition of "line" "point" and "straight" are defined in the theory. Doing an actual deduction in ZFC is very tedious; you can see here for the details of as well as the actual proof of 2+2=4 in ZFC.

So hopefully it is clear now why the relative assumption of an axiom in one case doesn't mean it is "without a proof" in all cases. But, this also sounds like it might end up in circular reasoning, so let me show how that is dispelled.

Obviously, if we say "well in one theory we have axioms A and B and those axioms prove C and D, and in another theory we have axioms C and D and those axioms prove A and B, so we have a perfect set of theories!" then we have circular reasoning. While those statements may be true, it would be dishonest to use them to justify each other. We need to pick one theory as a bedrock and have those axioms assumed in all cases, otherwise everything would be circular.

Philosophical Justifications for Axioms

If we agree that it is essential to have a bedrock of assumed axioms that will never be justified in our theory, how do we choose them? In mathematics, especially after the formalization that happened after the foundational crisis around the turn of the 20th century, philosophers, logicians, and mathematicians have spend considerable time discussing the axioms we eventually settled on (ZFC) and how we can justify what they say. Honestly, a lot of these ideas are not new and go back to the ancient Greeks.

A main idea is that the axioms are self evident, meaning that when we think about what they mean, they seem plainly obvious, even definitional. This is what Euclid thought of his postulates. There are translations of the Greek word "axioma (ἀξίωμα)" that include "self evident" in the definition.

Now, take for example the ZFC axiom of extensionality. This axiom states that if two sets have exactly the same elements then those two sets are equal to each other. This axiom, although it cannot be formally proven within ZFC, seems almost definitional to the idea of equality, doesn't it? It also is a perfect candidate for something that is self evidently true. Is it possible for two things to be made of exactly the same things and yet be different? It seems, at least to our rational intuition, that that cannot be true. A contemporary defense of the ZFC axioms, as pointed out in the comments, is Believing the Axioms by Penelope Maddy. In it, Maddy even argues that the axiom of extensionality seems to be the most definitional of all of the ZFC axioms.

The Validity of a Proof

In regards to your final question, the answer is no, there is no relationship between proof in mathematics and faith. Mathematics uses deductive reasoning, in the form of a formal proof system which are objects studied in proof theory. During a mathematical proof, you start with some statement and then apply a rule of inference to gain a new statement. In mathematics, we start with axioms and use our rules of inference to derive theorems, and often times we then can use those theorems to derive more theorems and so on. However, this method of inferring conclusions from premises has nothing to do with faith or with assumptions.

In proof theory, a set of rules of inference can have two important properties: validity and soundness. A valid argument, where argument means a specific instance of some premises and a conclusion, is one where "if the premises are true, then the conclusion must also be true." An example of a valid argument is:

All dogs are purple

Molly is a dog

Therefore Molly is purple

Now, if this argument is about the real world then we know that it is not true. However, it is a valid argument because the conclusion is assured to be true if the premises are true. Soundness is similar, except that sound arguments actually are true. For example:

All dogs are mammals

Molly is a dog

Therefore Molly is a mammal

Soundness is stronger than validity (notice that the sound example is also valid). The proof systems that are used in modern mathematics are not only valid, they are sound. It is precisely why we choose to use them, because it is a perfectly mechanical process to test whether or not a proof system has those properties. It has nothing to do with faith or assumption, those properties are things that can be shown. The only assumption we need to make is what axioms we pick, and there is a long history of philosophical arguments as to why the axioms we have picked are good axioms to believe in. The proof is the part that does not require assumptions; choosing axioms is the only step that does.

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    The first time I read this I was't sure if I agreed, but I think an important point that you treat carefully, is when you point out "faith as defined in this way" -- which I think addresses my worry well. It's not even clear if this is a good picture of what anyone really means by faith. – virmaior Jul 4 '17 at 11:27
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    I totally get that, I think that the point that I wanted to stress was just "yes there are assumptions being made and I don't want to hide that, but I don't think faith is the right word for it." Like with the example of the axiom of extensionality. If we say that we're only talking about sets and the elements are a set are the only things that differentiate sets, then saying "if two sets have the same elements they are the same set" that seems completely believable to me. I don't think that I'm using "faith" when I believe that statement, at least that isn't the word I would pick. – Not_Here Jul 4 '17 at 14:28
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    @Dennis I can't say that I see something wrong with that from a logical point of view, nothing jumps out at me right away, but I would disagree with that methodology for philosophical reasons. I don't think that's how we innately do math, especially not contemporary math, and I don't think that is the "correct" way that we should do it (obviously that's taking a personal stance in regards to the philosophy of math). What I mean is, if I were writing a paper in response to your claim I wouldn't argue against its coherence, I would argue that it isn't the right thing to do, if that makes sense. – Not_Here Jul 6 '17 at 2:37
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    Maybe I would make some sort of an analogy to the intended or standard model of a theory, and this being the standard or intended "proof theoretical schema" or something similar. But again, that is a philosophical argument, not a purely logical argument. Like, we can't have a purely syntactic argument for why a standard model is standard, assuming it has multiple isomorphic models, we can only have a philosophical one. If what you are suggesting is truly isomorphic, I would personally argue that it isn't standard or intended, whatever that means. But of course I could always be wrong. – Not_Here Jul 6 '17 at 2:44
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    @Not_Here no, I agree with you completely. I share that opinion. My only reason for raising the issue is that I'm not sure the virtues of the axiomatic methodology are truth-tracking virtues that give us reason to believe axiomatic theories are more likely to be true (or better justify our "faith" in them). I certainly find it both aesthetically and pragmatically abhorrent to work with complicated natural deduction systems -- metatheory is a nightmare. I'm just not sure that if "leaps of faith" are what you're concerned with, that axioms, but not proofs, require such "leaps". – Dennis Jul 6 '17 at 2:45
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The word "faith" is simply too loaded a term to use in any forum where the objective is open dialog because the term is identified more commonly by its misuse in a religious context as a term related to obedience to dogma. Nevertheless, your question's meaning is clear and the answer is fundamental to the use of rigorous systems of logic.

A far better term intended for this topic is "credence". The fundamental decision making model in the animal brain is Bayesian logic. That is, we learn from our experiences our personal truth on a matter. We each have a world view that is a collection of beliefs about things called credences that can be expressed as probability beforehand that I will accept as true the next thing I see. The more consistent our experiences regarding a matter, the strong our belief in that truth becomes. One example of a new truth is seldom enough to change our minds; but, it should increase my credence on the matter, thus making it more likely I will accept it next time. This system of reasoning is called abduction, as opposed to induction or deduction.

Now, man has known for a long time that abduction is fraught with pitfalls because of its very subjective and localized nature. So, early, most notably Aristotle and his contemporaries in the west, tried to develop "objective" systems of logic; in all cases, certain assumptions were necessary to proceed. Aristotle assumed that any logical and meaningful statement must either be true or false. Plane geometry assumes a well defines domain of an infinitely flat surface. Newton Physics assumes that time is a constant and proceeds forward in the same manner everywhere. Einstein assumed that light speed is constant in any frame of reference in order to derive Special Relatively.

So, axioms play an important role in even the most rigorous investigation based on inductive and deductive reasoning, and setting these axioms uses abduction. It is a general rule of thumb for most investigative diciplines that we not debate the answer before first debating the assumptions. Therefore, yes, you are correct that any "proof" or conclusion is tied to the assumption used in the derivation. Therefore, the assumptions are the first place we look for a system's boundaries or domain of applicability.

It is not, however, necessary for one to "believe" an assumption or axiom in order to play the game of "what if?". In fact, it is often the case that starting from an assumption is contrary to common belief results in a paradigm shift in man's understanding of nature and reality.

  • I'm not entirely certain I understand the statement more commonly by its misuse in a religious context as a term related to obedience to dogma. Are you saying (1) this is not how it is used in religious dialogue, (2) this is how it is always used in religion and this is somehow a misuse (as if there's a more original non-religious meaning), or (3) this is how it is sometimes used in religion, and this is a misuse vis-a-vis ???. – virmaior Jul 4 '17 at 11:29
  • Ah, never use the terms always or never ;) I thought your use of the word faith was fine and contextually correct; I just suggested another term because many in the scientific community have been burned by "the faithful." If I hold 100% credence in any belief, whether it be scientific or religious, I can't even see contrary evidence let alone judge it. This is self-imposed ignorance, not faith; It is irrational and not a very good survival strategy. In this small space I might easily unintentionally offend or anger you. Also, not good topic for this stack. We can meet elsewhere. – Ron Gordon Jul 4 '17 at 14:08
  • Virmaior, I am new to the stacks, and I see with some embarrassment that my explanations for credence must seem sophomoric to you. I just earned chat privileges. If you wish, we could meet there. I don't know how and I don't know the rules of engagement. So, if you agree, I asked that you take the lead. Ron – Ron Gordon Jul 4 '17 at 14:29
  • That's fine. I'm not actually doubting you per se. I just wasn't sure what you meant with your statement there. Apart from that, I think your answer makes a lot of sense. – virmaior Jul 4 '17 at 15:42

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