/sci/ - Science & Math

>>3685306
Nope. It's all arbitrary.

>>3685306
Based on anything?
Wtf do you mean?
It's math dumbshit.

Axioms are unprovable by fucking definition. Any advanced systems of logics will lead back to axioms, WHICH CAN NEVER BE PROVED! (else they wouldn't be axioms).

It works kind of like this pic.

The real shocker is that those arbitrary axioms work so well in describing the world around us on even the tiniest scales.

>>3685317
Not exactly. There are many sets of axioms that at least produce results that match reality to some degree. They can be completely arbitrary but often they aren't.

The cool thing about the axioms of math is that the conclusions you can draw from them make sense with reality.

I took maths up to A-level and half way through MEng.

What are the axioms of mathematics? Why?

>>3685342
> thinking axioms are arbitrary
> not realising that axioms are chosen because they are seen to correspond to the real world
> complete moron
> probably one of those 'can't be bothered with philosophy' idiots

>>3685349
bmup?

>>3685349
Well geez, there is a shitload of them. Maybe a bunch for each "fields"
(Euclide's, Peano's...)

>>3685349
an axiom is a proposition considered to be self-evident

>>3685359
>axioms are chosen because they are seen to correspond to the real world
Sometimes. Some mathematicians don't give a fuck however. But even then there might be a future technology that does correspond to what they did theoretically and then their work becomes practical.

>>3685359

What you observe doesn't necessarily match what I observe and yet we both agree that the set of axioms matches what we see around us.

The set of axioms is arbitrary because there could be a different set of axioms that accomplishes the same purpose of describing the world around us. If the mathematicians that formulated those axioms had instead taken up cooking and someone else was tasked with creating a set of axioms it would most probably be different than what we have today.

And failosophy is for idiots too scurred of real science.

>>3685374
I think they've given up on uniting them.

>>3685375
More like something clear or simple enough for the major part of users to agree with.
There are often individuals that choose not to use a particuliar axiom, which leads to a whole new theory.

There's a common misconception that mathematics can only exist founded off of the axioms we've chosen. Fact of the matter is that we could have just as easily chosen different items as axioms and the rest of the math would've structured itself the same way. The only thing making the axioms we've chosen special is that they're more or less the most fundamental axioms we could decide on (that is to say that we wanted to build the rest of the math on the fewest possible axioms). There is really no special reason to restrict mathematics to just axiomatic systems. It just works out a lot easier and allows us to really minimize doubt and prove things with absolute certainty. It does however also create some pitfalls, like we've seen with formalism and the incompleteness theorems (formalism being a mathematical system founded on logic, and the incompleteness theorems proving that such a system could be either complete or consistent, but not both).

>>3685372
depends on your philosophy.
axioms in general will correspond intuitively to some class of objects.
for instance a + b = b + a obviously applies to all objects that arithmetic works for.

Well could you describe a few axioms to a non-maths student, so I know what we're dealing with?

>>3685397
Here, if you like natural numbers read this:
http://en.wikipedia.org/wiki/Peano_axioms

>>3685381
protip: don't constantly talk about how philosophy is pointless and trivial if you're hilariously incompetent at it. just look at this shit:

> What you observe doesn't necessarily match what I observe

> implying there is no objective reality with consistent laws
> in /sci/

idiot.

>>3685342
That's because mathematics is based on conventional quantities. Reality can be described based on any constant convention.

>>3685397
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html

Postulates and Common Notions.

>>3685419
>Euclid's definitions

>>3685379
> Sometimes. Some mathematicians don't give a fuck however.

not really true. i've yet to see a set of axioms that don't describe some kind of object; this is a mathematician speaking.

>>3685397
http://en.wikipedia.org/wiki/Euclid%27s_postulates#Axioms

It is important to note that for a loooooong time many of the truths of geometry had already been known. Euclid didn't contribute much in the way of new theorems, instead what he did was boil down all of [euclidean] geometry to just 5 axioms (for a long time believed to be 4 because it was thought that the 5th could possibly be proven using the other 4) upon which the rest can all be constructed.

>>3685397
For instance, you "decide", in Euclidian Geometry, that a single line can be drawn through 2 points.
You cannot prove this. It is used as an axiom.

Some people decide not to "believe" in this. They can create another axiom, or you could say that the decide to agree with another idea, for instance that you can draw 2 lines between 2 points. It can then be expanded inton another theory.

You may be able to "prove" or disprove that axioms hold in "reality", however logically you just assume that they are true.

>>3685317
This. You are more or less right, OP. But one can argue that pretty much all scientific endeavour is axiomatic. It's a frail argument though; at some point, after something is proven time and time again, we can make a valid assumption that we're onto something.

>>3685439
I mean if your math even is analogous to anything physical.

>>3685425
Line and points are abstractions dude. Where is the "correspondence with reality"?

>>3685342
>>3685359

Mathematics does not describe the real world by its purpose, it's only a side effect. In other words, the mathematics of the real world are just a tiny subset of what mathematics encompasses. Similarly if the physical laws were to all be drastically changed, the mathematics would still be the same.

>>3685405
They seem so intuitive, it's unbelievable that they have to be written down. I guess the only sets I have to care about are R R2 R3 and C....

It's even strange to see that natural numbers can't simply be stated with:
0 is in N.
1 is in N
If k is in N, k+1 is in N.

>>3685423

I'm going to go out on a limb and guess that you're just a highschool teacher.

So many autistic idiots in this thread.
Axioms dont need any proof.

>>3685425
> You cannot prove this. It is used as an axiom.
> not understanding empiricism
> in 2011

look at the world. the axiom always seems like a good approximation of 3D space. this is how scientific truth works.

>>3685454
that definition is completely circular. what are 0 and 1?

>>3685406

How do you know what those consistent laws are without observing them how objects interact? We conceive of reality what we see.

You "objective reality" is based on observation. Enough observers agree that reality is behaving in a certain way and we devise axioms to describe it.

Failosophy is just a bunch of question-dodging. It's not a hard science and it creates delusional people with angry panties.

Go back to /lit/.

>>3685470
Conventions, like all numbers, signs and symbols.

>>3685464

Go learn some hypergeometry, higher dimensional spaces (like quaternion space), and other non-euclidean spaces first plz.

Mathematics is not concerned with the real world. The real world is just an imperfect approximation of the math.

>>3685448
> Mathematics does not describe the real world by its purpose, it's only a side effect.

...no.

brains are a product of the universe. the rules of the universe are inescapably hard wired into what we think about. we would have no conception of number without distinct solid objects, for example. if you plan to apply it or not, this is where mathematics comes from.

>>3685483
No, I mean that you can't just say:
"0 is in N"
"1 is in N"
with no reference to their properties

>>3685500

>implying mathematics is just about numbers
>implying mathematics are bound to this universe

Perhaps you may want to read into the three foundations of mathematics first (formalism, platonism, intuitionism) before you go off spouting bullshit.

>mfw there's one idiot failosopher in the thread arguing against all logic and reasoning put forth by the sane scientists and posters on /sci/

>>3685469
> observers agree that reality is behaving in a certain way and we devise axioms to describe it.
> exactly what i was arguing for and you were arguing against
> confirmed for epic retard

>>3685513
>implying mathematics is just about numbers
"for example". english motherfucker; do you speak it?

> Perhaps you may want to read into the three foundations of mathematics first (formalism, platonism, intuitionism) before you go off spouting bullshit.
> implying professional mathematicians base their ideas on platonism
> implying any serious academic still believes in platonism
> in 2011
bro, you clearly have no clue what you're talking about. it's tragically hilarious; please stop.

>>3685306
Its better than something based on easily disproven axioms.

>I'm looking at you, Economics.

>>3685530

LMAO. You forgot which post you replied to first, didn't you, you adorable aspie son of a bitch.

I claimed the axioms were arbitrary, not that they didn't describe what we agree is reality. That set of axioms could be replaced by another arbitrary set of axioms that describes it just as well or better or worse.

That was the OP's point, you retard. Failosophy strikes again.

Read this post >>3685389 for a better phrasing of the argument since you obviously can't read the posts you're replying to.

>>3685547

The foundation of Mathematics version of Platonism, not your fail philosophy platonism.

Most mathematicians now are Platonists, then Formalists, and a significantly smaller number are intuitionists.

Formalists were the biggest group back in the day, but then godel (a platonist mathematician) developed the two incompleteness theorems which basically disproved formalism. Now it's just a bunch of people who are either in denial, or scrambling to find some loophole that allows formalism to continue.

>>3685500
>the rules of the universe are inescapably hard wired into what we think about.

can you prove that, faggot? If you're going to make shit up, at least try to make it believable

>>3685568
> I claimed the axioms were arbitrary, not that they didn't describe what we agree is reality. That set of axioms could be replaced by another arbitrary set of axioms that describes it just as well or better or worse.

probs the least coherent thing i'm yet to read on /sci/, grats.

>>3685513
What's your proof of this? Math dogma fail

>>3685547

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

fail more

>>3685608
> the brain has evolved to think about this universe
> "not believable"

LOL

Inb4 Solipsism

>>3685611

He's saying that if the axioms we've chosen could be derived from a different (possibly larger) set of axioms, then we may as well have chosen those instead and we'd have the exact same mathematical system we have now.

In this sense the axioms are arbitrary while the math is not.

I'm pretty sure math started out as a tool to describe and understand occurences in reality. A classic example is the counting of chickens, or the length of wood needed to build a fence. One thing leads to another and now were mostly dealing with math that is used to do more math.

>>3685615
wtf is this shit?

Math: the language of certainty.

>>3685627

Don't sweat it bro. He understood it just fine. He's just buttsore because he's posted over a dozen times in this thread arguing against what everyone else already proved beyond a reasonable doubt to be true.

He's an aspie failosopher so he'll get over it pretty quickly.

>>3685627
that's not arbitrary at all. if your set of axioms is equivalent to, or implied by, some other set, how on earth can they be called arbitrary?

> mfw this guy was complaining that philosophy is semantic nonsense
> and he's using words backwards

>>3685637

Mathematicians trying to pin down what math really is.

Logicism is more or less a fake foundation, just philosopher fags trying to pose as mathematicians to seem relevant. The only other foundation that founded mathematics on logic was formalism, but that's been largely disproved. The consensus now is that philosophy as a whole is largely uninvolved in mathematics or is at best a subset of mathematics (philosophy based on mathematics).

The other way around creates logical paradoxes that allow people to make statements like 2=1

>>3685661

Choosing them is arbitrary. Think of the mathematics as a hugely parallel network of theorems. Some depend on others, many depend on more than one. None really depends on just itself. What axiomatizing the mathematics does is go through said network and pick out a few "chosen" theorems upon which all of the others can be derived in a system where one states "if I assume that this small handful of statements are true, then I can prove that all of these other statements are also true". Not everyone agrees with axiomatization, though nowadays the applications have become boundless by studying axiomatic systems and tweaking said axioms to produce new results and correlate said results to the axioms in deeper ways. This is what we see now in abstract algebra (also sometimes called modern algebra). It is the study of all possible algebraic systems built on all possible sets of axioms and operations. It includes set theory algebra, matrix algebra, lie algebra, etc.. There are theoretically an infinite number of algebraic systems out there, and most of them do not relate to reality in any way. Currently there are probably around 200-300 algebraic systems being actively studied, most but not all have real world applications (which is why people have chosen to study them).

>>3685707
> which is why people have chosen to study them
> Mathematicians actively choosing to study something interesting and useless instead of something interesting and useful

>>3685734
Prove it

Axioms don't need to be proven you retard.

>>3685723

Just because an algebraic system has no real world applications does not mean that it is not useful.

The goal of abstract algebra is to find ways to categorize and deduce properties of ALL algebraic systems, even those that have not been conceived. An advancement in all of abstract algebra by far trumps any advancement in a single application of an algebraic system.

>>3685747
> Just because an algebraic system has no real world applications does not mean that it is not useful.

intredasting...

math is based on axioms, but for the most part challenging the axioms would be like being the idiot who, when his friend says "check out that cool car", responds by saying "how do you know theres REALLY a car there?"

>>3685738
Because maths is the study of logically deriving theorems and results once axioms have been decided on.

There doesn't even exist a context within mathematics with which to say whether axioms are true or not. Those labels are meaningless.

>>3685747

>Just because an algebraic system has no real world applications does not mean that it is not useful.

>The goal of abstract algebra is to find ways to categorize and deduce properties of ALL algebraic systems

Note to self dear Anons: Algebra is useful because it can categorize and deduce its own properties.

>>3685768
we can agree though that if the axioms correspond to observation then so will the math; if the axioms do not correspond to anything in reality than neither will the math.

>>3685778
False. Go learn some Goedel's Incompleteness Theorems.

>>3685778
>categorize and deduce its own properties.
>Its own properties referring to the properties of certain algebraic systems
>many of which have important practical applications

I think one day we'll be able to prove axioms. Remember, at one point we thought there was nothing smaller than atoms, but then we found out about protons and neutrons. I think we'll do the same with axioms and find principles of logic even deeper than them

>>3685784

Anon: Dear Tripnon but why should i learn Algebra?
Tripnon: Because learn some Goedel's Incompleteness Theorems.
Anon: ...

>>3685794
Again, no. Go see Goedel's Incompleteness Theorems.

"Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true ... If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

-Bertrand Russell

What the faggot wants to say is that math is not concerned with absolute truth. It's only concerned in relations between propositions, that is it assumes P is true and see what else must be true in that case, then it assumes P is false and see what happens.

>>3685801
You claimed that algebra can "deduce its own properties". That claim is false. Goedel's Incompleteness Theorems prove that claim to be false.

>unproved
>axiom
pick one

>>3685807

I think that anon misunderstood in thinking that said deductions were done from within an algebraic system.

They are not, this does not violate godel's incompleteness theorems.

>>3685807

If you were more literate and less autistic you would have realized that i was quoting and mocking another anon.
Your interference is entirely unnecessary.

>>3685820
Agreed. However, he very clearly said it can deduce /its own/ properties, which means you're giving too much benefit of the doubt here.

>>3685826
Yes, I'm obviously autistic because Poe's Law.

Protip: Sarcasm doesn't carry well in text.

It's not based on anything - it's a network of pure analytical reasoning. The fact we pick certain nodes in the network as axioms from which to deduce the rest is by tradition and entirely artificial.

pic related - is "analytical" not a word?

>>3685835

I always imagine the nodes as being green and red circles for true and false respectively, and being linked by one or two sided arrows that "carry" the green over to the next node. You just have to paint a couple of nodes green and the truth spreads to all the propositions which are consequence of the ones you chose. Makes one simple picture of what an axiomatic system is.