
How is a system of axioms different from a system of beliefs? - brudgers
http://math.stackexchange.com/questions/182303/how-is-a-system-of-axioms-different-from-a-system-of-beliefs/182308#182308
======
ot
This question gets asked a lot, in particular by math students, and I think
that the answer that is usually given, which is the same as many answers in
this page, is overly simplistic: we don't care about ontological truth of our
axioms (and, by the way, of the deduction scheme), we just define a set of
axioms and define true as _provable_. This is the formalist view of
mathematics, which was the dominant view at the beginning of the 20th century.

This reduces math to a meaningless mind game, as acknowledged by Hilbert
himself, one of the major proponents of this view: "Mathematics is a game
played according to certain simple rules with meaningless marks on paper".

Unfortunately, this sweeps under the rug important questions, such as "if I
prove the correctness of a an algorithm, how does this increase my confidence
that the algorithm will actually work?", or "if I compute the hypotenuse of a
triangle with Pythagoras theorem, then cut a piece of wood and measure it,
will they match?"

This is why ultimately there is some belief involved. From Wikipedia [1]: "In
practice, most mathematicians either do not work from axiomatic systems, or if
they do, _do not doubt_ the consistency of ZFC, generally their preferred
axiomatic system" (emphasis mine).

To work with with a set of axioms, we must _believe_ that that set is both
logically consistent, and that in some way its interpretation is consistent
with reality. The first question cannot be answered for the interesting sets
of axioms, thanks to Godel, and the second is not even well-defined.

For this and many other reasons, philosophy of mathematics has mostly moved on
from the logicist/formalist view [2], but as most philosophical problems,
there probably will never be a conclusive answer.

[1]
[http://en.wikipedia.org/wiki/Foundations_of_mathematics#Part...](http://en.wikipedia.org/wiki/Foundations_of_mathematics#Partial_resolution_of_the_crisis)

[2]
[http://en.wikipedia.org/wiki/Philosophy_of_mathematics](http://en.wikipedia.org/wiki/Philosophy_of_mathematics)

~~~
Dylan16807
Completeness is overrated. Why does it matter that a system can't prove every
possible statement you can write in it? It's not going to give you a wrong
answer, it's just going to fail. Why is that particular lack of capability
more important than the millions of statements you can't formalize in the
first place? Take consistency and build something with it.

P.S. Using a theory to prove its own consistency would be way more suspicious
than using a more powerful theory to prove it. "The Bible is true because the
Bible says so."

~~~
yodsanklai
On the other hand, consistency alone isn't really useful!

Let suppose I design a new proof system to prove arithmetic propositions.
Unfortunately, my system is such that it can't produce any proposition. That
makes it consistent but useless.

~~~
Dylan16807
That's why you look at completeness in terms of coverage, not as a binary.

------
fxn
Axioms are not beliefs. Axioms are a priori and you formally derive stuff from
them using particular logic rules.

Generally speaking, nobody claims that the axioms of set theory are true in
any ordinary sense of the word true. Does it makes sense in the real world to
assert the existence of an infinite set, and have different sizes of
infinities as a consequence? It is irrelevant, you assert the existence of an
infinite set because it is practical from a pure mathematical point of view,
because you want to model infinite sets like the natural numbers (an
abstraction, do they exist? it doesn't matter to mathematicians generally
speaking).

Mathematics no longer pretends to describe what is true, what holds in our
reality. That idea was abandoned some time ago. A canonical example were non-
euclidean geometries, that were studied in the XIXth century for the sake of
it. They had applications later, but the motivation for their study and
changing Euclid's axioms was formal.

~~~
cwzwarich
> Generally speaking, nobody claims that the axioms of set theory are true in
> any ordinary sense of the word true.

Mathematical platonism is a pretty widely held belief, at least amongst pure
mathematicians. From Hardy's A Mathematician's Apology,

"I believe that mathematical reality lies outside us, that our function is to
discover or observe it, and that the theorems which we prove, and which we
describe grandiloquently as our 'creations', are simply the notes of our
observations."

A lot of set theorists will speculate about the truth of axioms beyond the
standard axioms of ZFC, e.g. large cardinal hypotheses or projective
determinacy.

------
azov
There's an infinite variety of axiomatic systems, most of them not much
different from, say, a set of beliefs in Santa Claus and magical elves living
at the North Pole.

However the axiomatic systems we happen to use most have one important
property: they are _useful for acquiring new knowledge via scientific method_.
I.e. they serve as a foundation for theories that make correct predictions as
verified by experiment.

That's the main difference.

~~~
vinchuco
A story:

"There's two scientists on lab coats observing a spider. They shout JUMP! and
the spider jumps.

They proceed to remove one of it's legs and repeat the procedure. They shout
JUMP! and the spider jumps...

After some time of repeating the procedure the poor spider has no legs. They
shout JUMP! and it doesn't do a thing.

The scientists then come to verify their useful hypothesis: ..spiders use
their legs to listen..."

~~~
jlgreco
Cute story, but don't mistake it for a second for any sort of casual
indictment of the scientific process. ...the story does not even feature the
scientific process in the first place (which goes a long way to explain why we
find it silly).

------
javert
This is pure Platonism. I reject Platonism in modern mathematics.

Mathematics should, first, be used to describe the real world. (That is what
scientists like Newton were using it for.)

To do that, you don't need an axioms. Nor do you need beliefs. You only need
observation of reality. It starts with the concept of a "unit," which can be
added or subtracted from other "units." There is, of course, some prerequisite
_philosophy_. ("Reality, you say?" "Why, yes, actually...").

Secondly, mathematics can deal with axioms and "games" and "universes" (to
take words from the StackExchange post). But all of mathematics should not be
subordinated to such a framework.

I realize my argument is not "obvious." Fully establishing it would take a lot
of work, and even then it would be widely rejected by the status quo Platonic
mathematics establishment. I'm not claiming to do that in this comment. I'm
just putting this alternative perspective out there for anyone who is
interested.

~~~
EliRivers
_Mathematics should, first, be used to describe the real world._

If by "real world" you mean the things and events we can directly witness,
that seems to be deliberately crippling mathematics from the start and
limiting it to things we already know about.

~~~
javert
Depends on what you mean by "directly witness," but that's neither here nor
there.

What I am proposing would not cripple mathematics, it would simply replace
"axioms" with _facts_ (such as, for a simple one, that adding one unit to
another results in having two units). You can then get arithematic, algebra,
calculus, and on and on.

If people want to go off and define axioms for made-up worlds that do not
necessarily correspond to reality and then go do math with them, that's fine.
In fact, I would encourage it.

But we should not be approaching _all of mathematics_ in that way, and that is
my point.

Math has the same basis in perceptual concretes (i.e. "real things") as
physics, biology, chemistry, etc.

~~~
gjulianm
The ZFC axiom set [1] doesn't define a made-up world. Actually, once you
understand the axioms (they're somewhat difficult in notation and definitions,
but the ideas behind are understandable) they make sense. There are some
axioms made to avoid paradoxes (notably ZFC3 [2]), but the majority describe
how we understand the world. It'd be difficult to imagine a world where two
sets aren't equal even when they have the same elements, or where you couldn't
construct a set that contains all elements of two other sets.

How's this different to a set of _facts_ as you say, like one unit + one unit
is two units? It's not. I'm not sure of this, but I'd say you could construct
a set of axioms based on those _facts_ that is compatible with the ZFC axiom
set.

So, why choose ZFC? Because it's simple to define (how'd you formally define
what is a unit, and what does the + operation mean?) and because it's more or
less consistent with our world. We can extrapolate the things we find while
doing mathematics to the real world. And, sometimes, we can discover things of
the real world based on findings on things that don't _exist_. Complex
numbers, for example: they don't exist, but they're really useful to define
and explain electrical experiments [3].

We approach _all of mathematics_ like this because its easier to study
everything within the same system. And it's important that we keep a
consistent underlying system so we can be sure that everything we discover
based on the set of axioms and rules we've defined is _true_ in a mathematical
sense; and then those discoveries could describe our world. Wouldn't we have a
consistent system, we could end up with contradictions. If we extracted
mathematical concepts from the real world, we wouldn't be sure if a
_discovery_ we make from those concepts can be applied to the real world.

TL;DR: Our set of axioms describes properly the real world, is simple and
consistent enough and that enables us to infer new ideas that can be
applicable to the real world too.

[1]
[http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_th...](http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)
[2]
[http://en.wikipedia.org/wiki/Axiom_schema_of_specification](http://en.wikipedia.org/wiki/Axiom_schema_of_specification)
[3]
[http://en.wikipedia.org/wiki/Complex_number#Electromagnetism...](http://en.wikipedia.org/wiki/Complex_number#Electromagnetism_and_electrical_engineering)

~~~
EliRivers
_Complex numbers, for example: they don 't exist, but they're really useful to
define and explain electrical experiments_

At risk of turning this into a discussion of what "exist" means, I reckon they
do exist :p If they didn't exist, how could we possibly use them, and use them
so effectively and consistently?

Of course, if a number "existing" means "I can see this many number of apples"
then sure, it doesn't "exist". Yes, this is definitely becoming a discussion
of what the word "exist" means applied to number. I'll stop. :)

~~~
gjulianm
Well, yes, I was talking about exist in the sense "I can't count _i_ apples",
and like you say better stop here because this can get infinitely long :)

------
fela
You don't blindly believe axioms, it is just that certain sets of axioms
happen to allow us to make useful predictions about the world. While you could
have philosophical discussions about weather certain axioms are true (whatever
that means), and what their exact relationship with the world is, you don't
need to do this to see that axioms are useful to arrive at mathematical
conclusions, which then can be used to make predictions about the world.

Why do we believe that if we take 1111 apples and then add 2222 more we will
get 3333 of them? Because we have seen from experience that a certain way of
applying mathematics to the real world gives great results. The same for
conclusions following known axioms, they seem to generally work well for
describing our world.

------
fela
You don't blindly believe axioms, it is just that certain sets of axioms
happen to allow us to make useful predictions about the world. While you could
have philosophical discussions about weather certain axioms are true (whatever
that means), and what their exact relationship with the world is, you don't
need to do this to see that axioms are useful to arrive at mathematical
conclusions, which then can be used to make predictions about the world.

Why do we believe that if we take 1111 apples and then add 2222 more we will
get 3333? Because we have seen from experience that a certain way of applying
mathematics to the real world gives great results. The same for conclusions
following known axioms, they seem to generally work well for describing our
world.

~~~
VLM
A good post although I suggest rephrasing "useful" to something like
"experimentally falsifiable". With a pretty wide definition of
"experimentally" and "world", especially for non-applied math.

If you have no way to do a falsifiable test, and can't just substitute in a X
and do abstract analysis of X instead of worrying about X, then its just a
belief.

I'd be careful categorizing stuff eternally as beliefs not axioms. Blindly
only permitting geometry to be Euclidean meant some exciting non-euclidean
results couldn't happen until it was "allowed" to be thought about somewhat
recently (well, a century or two ago...). Also physicists have this amazing
ability to turn pure math into applied math, at least over a long enough
historical scale. And physicists have a pretty good ability at coming up with
crazy experimental methods to test, look at the last 75 yrs or so of quantum
physics... so let me get this straight, you do what with two geiger counters
and a truly giant magnet and some radioactive atoms or what? Or shine a light
beam or ion beam thru that weird magnet?

------
jkbyc
This sort of thing is also studied as part of
[http://en.wikipedia.org/wiki/Epistemology](http://en.wikipedia.org/wiki/Epistemology)
and it may be of interest that it is studied also as part of computer science
for example as
[http://en.wikipedia.org/wiki/Belief_revision](http://en.wikipedia.org/wiki/Belief_revision)
which is in turn closely related to
[http://en.wikipedia.org/wiki/Reason_maintenance](http://en.wikipedia.org/wiki/Reason_maintenance)
which used to be the basis for general problem solvers. By the way, view
maintenance in relational databases could be seen as a subset of reason
maintenance.

Note that belief revision and various kinds of paraconsistent logics in a way
allow to work around the classical requirements for consistency.

Another really interesting related area is
[http://en.wikipedia.org/wiki/Modal_logic](http://en.wikipedia.org/wiki/Modal_logic)
which allows you to reason about statements such as "I know that you know that
I know that the sky is blue." and that can be used to analyze the coordinated
attack problem
([http://en.wikipedia.org/wiki/Two_Generals'_Problem](http://en.wikipedia.org/wiki/Two_Generals'_Problem))
which is very much related to the
[http://en.wikipedia.org/wiki/Byzantine_fault_tolerance](http://en.wikipedia.org/wiki/Byzantine_fault_tolerance)
of distributed systems.

------
bladedtoys
There maybe some axioms that are analogues to rules of a game and so are
neither true nor false.

But in practice, the bulk axioms are chosen and retained because the inventor
thinks they or their consequences have some mapping to reality. This mapping
is tested through applying the resultant math to some field and getting
predictive success. In this regard the bulk of axioms are the same as any
other beliefs except they are not arbitrary beliefs, and they are useful or
believed to have the potential of usefulness.

And part of the problem is merely semantic. The word "belief" differs from the
term "axiom" in other critical ways: "belief" regularly involves a call to
action and includes concept that do not go though any tests for consistency
let alone predictive success. If one asks "what is the difference between an
assumption and an axiom" there is a lot less excitement.

Most interesting, I think, is the idea that basic axioms should be different
than assumptions at all. That they can be definitively decided or the part of
reality they map to is real and can be found.

I do not see why we would necessarily have evolved the mental equipment to
work out every law of nature. No other animal has that, and while we clearly
have better mental capacity I can't see why we would have evolved _every_
mental capacity.

------
logicallee
The major difference is that nothing requires that your set of beliefs be
consistent: however, if a given set of axioms allows you to prove and disprove
some assertion _P_ , you pretty much can't use that set of axioms anymore!
Well, you can, but rather than the normal huge castles made of sand that
follow from any reasonably strong set of axioms, you will end up with a small
puddle of mud that's useless.

In this sense, a simple discovery (of an axiomatic system's inconsistency)
would essentially cause that set of axioms never to be used again.

Can you imagine that a simple logical proof would immediately cause the Pope
to abandon making certain statements together?

For example check out this post:

[http://www.religiousforums.com/forum/1993062-post3.html](http://www.religiousforums.com/forum/1993062-post3.html)

Suppose a religious scholar produces an argument that the universe described
by,

"Jesus Christ, the only Son of God" \+ "who was born of the Virgin Mary" \+
"his kingdom will have no end" is an inconsistent Universe.

It won't suddenly be the case that nobody will mention these three things in
the same axiomatic set again, just because they're (through some rigorous
proof) inconsistent.

That's a huge, huge difference. It's like the difference between chess and
Shakespearean criticism!

------
mk270
A couple of important distinctions between axioms and beliefs:

1) axioms are assumed to be logically consistent with each other, at least
when the word "axiom" is understood strictly; it's possible that someone may
believe two things that are inconsistent with each other

2) belief is an involuntary mental acceptance that a particular claim is true.
You can't _make_ yourself believe something that you know to be false

------
bane
Axioms are definitions you set before you start to derive the rest of the
system. If you want to, you can test the axioms and the derivations against
reality and you have science.

Beliefs are definitions you think somebody else set that you can't change
before you start to derive the rest of the system. In most cases, you are not
permitted to test beliefs against reality.

------
nraynaud
I stopped reading when he implied that projective geometry is a wonderful
game. My last party ended up with me defeated and crying.

------
vinchuco
Axioms are building blocks. Beliefs are the idols you make with or without
building blocks.

------
lotsofpulp
Assumption (or axiom) + ego = belief

You don't get your feelings hurt when an assumption turns out to be wrong. But
if you were to be betrayed by someone you believed (same as assumed they were
telling you the truth or had your interests in mind), it would hurt your
feelings because you had invested some of your ego with that assumption. Same
for those with religious beliefs.

I'm using axiom and assumption interchangeably, as I don't see what the
difference between the two is. However, assumption is a much simpler word and
means pretty much the same thing to everyone, so it's easier to use. I guess
axiom is something like a generally accepted assumption.

------
gosub
I also religiously believe that a flush is better than a pair in a game of
poker.

~~~
saraid216
That isn't a belief with a quality of religiousness; it's exactly the same as
saying you religiously believe 2 is greater than 1.

------
samatman
A reality, often unacknowledged, is that mathematicians proceed largely on the
basis of elegance. As Achilles and the Tortoise shows us, basic logic has
built-in points of infinite regress, and the only way past them is to agree
that the logic is sound.

Mathematicians who must choose between two sets of axioms will pick the one
they consider most elegant, every time. The fact that this is ultimately an
aesthetic judgement is clear: Erdős Pál, arguably the finest mathematician of
the latter half of the 20th century, described proofs that tickled his
sensibilities as "Pages from the Book".

------
smnrchrds
The question is about comparing axioms with belief and religion; so my
question is kind of off-topic, but I have to ask: How about philosophy? How
does proving a philosophical theorem start? Is it by an axiom or something
else? To put it another way, what is the philosophical equivalent of
scientific method?

~~~
gbog
Descartes and Spinoza tried to apply scientific method to philosophy.

~~~
saraid216
[http://plato.stanford.edu/entries/peirce/](http://plato.stanford.edu/entries/peirce/)

------
eriksank
In my opinion, if the set of beliefs is orthogonal and irrefutable, it is a
set of axioms. Orthogonal: It is not possible to derive one belief from
another or a combination of other beliefs. Irrefutable: Counterexamples have
not been found or counterexamples can simply not ever be found. It is obvious,
however, that orthogonal and irrefutable do not mean "true". On the contrary,
"true" means that a statement can be derived from the axioms, while the axioms
by definition themselves cannot. Axiomatic systems are simply a form of
rigorous and highly systematic beliefs, but not necessarily internally
consistent nor necessarily consistent with reality.

~~~
gjulianm
In the same sense that an axiom can't be proved, it can neither be refuted. I
think you mean they're consistent: that you can't infer a paradox or a
contradiction from your set of axioms.

