Maths - invented or discovered? Watch

Robob
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(Original post by Tut.exe)
well if you can answer why can you model the universe using mathematics? and if it proves that mathematics can only point to an approximation then i'd stop believing it is.

Well yeah i know its ironic that at the moment it only shows an approximation.
Because the universe obeys some basic rules. These are then used to make logical deductions about what will happen. Given that maths is based on logic it is a very good descriptive tool.

This does not however mean that it exists beyond human concepts. A particle doesn't integrate the field it passes through to work out how fast it will move (for example).

Ironic? What?
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Swayum
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(Original post by Kolya)
Who says we need Maths to do Science, though? What about science which is based on a language other than maths? Couldn't Maths be a sufficient but unnecessary tool to do science? :wink2:
No one. Maths is a language of science, but not the only one. Much like English is one language for humans, but we could certainly do without it.
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Kolya
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To be honest, a thread with philosophy of mathematics and Godel's first incompleteness theorem is a little too much for me. ;no; Can't we go back to helping with C2 questions?
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Tut.exe
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(Original post by Robob)
Because the universe obeys some basic rules. These are then used to make logical deductions about what will happen. Given that maths is based on logic it is a very good descriptive tool.

This does not however mean that it exists beyond human concepts. A particle doesn't integrate the field it passes through to work out how fast it will move (for example).

Ironic? What?
no it doesnt work it out, it obeys the 'rules'. A mountain doesn't work out it's shape, it becomes what it is because of its properties and natural causes, the earth revolves around the sun not because it works out how and why it should but because of the relationship of the sun and the earth. what are you trying to point out with your particles?


so far, what people have known is pretty much generalised as an approximation, yet.
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sdt
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(Original post by generalebriety)
I wouldn't count that as maths, personally. Not on an "I'm too good for arithmetic" level, more because I don't count fixed quantities (basically, mensuration) as maths. Perhaps if you were manipulating these quantities somehow...
There are a lot of attempts to naturally derive those constants, supposing even one was correct then surely that would prove it?

Your own example of acceleration/velocity and differentiation, it's discovery in every way, shape or form.

Even simpler is the fact that these quantities just exist, regardless of what we do. Define c as 1, alpha=1/137. Define c as 3x10^8, alpha is still 1/137. The ratios of various particle masses and force strengths, they exist regardless of what framework we describe them in. So you can just reduce the question to "if a tree collapses and nobody is around to hear it..."

(Original post by Tut.exe)
no it doesnt work it out, it obeys the 'rules'. A mountain doesn't work out it's shape, it becomes what it is because of its properties and natural causes, the earth revolves around the sun not because it works out how and why it should but because of the relationship of the sun and the earth. what are you trying to point out with your particles?


so far, what people have known is pretty much generalised as an approximation, yet.
You would actually be wrong there, a mountain does "work out" its shape and the earth does "work out" so and so. At least according to decoherence theory (of which I know only that :ninja:)
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generalebriety
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(Original post by sdt)
There are a lot of attempts to naturally derive those constants, supposing even one was correct then surely that would prove it?
Sure, and supposing there was even one counterexample to Fermat's Last Theorem that would disprove it. Can you come up with a successful attempt to derive these constants without maths, please? :p:

(Original post by sdt)
Even simpler is the fact that these quantities just exist, regardless of what we do. Define c as 1, alpha=1/137. Define c as 3x10^8, alpha is still 1/137. The ratios of various particle masses and force strengths, they exist regardless of what framework we describe them in. So you can just reduce the question to "if a tree collapses and nobody is around to hear it..."
Yeah. I'm saying that alpha equalling 1/137 isn't really "maths". It's just mensuration.
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generalebriety
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(Original post by sdt)
http://www.physicsforums.com/showthread.php?t=117787


\frac{(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau})^2}

{m_e+m_\mu+m_\tau} = \frac{3}{2}

Still menstruation?
Yeah, but, I don't know any physics, so that's just meaningless words to me.

(Original post by sdt)
I mean, it is more of a losing battle if you're going to define "maths" as the opposite thing each time...
Don't know what you're on about. :confused:
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sdt
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Actually I thought those quantities were dimensionless for a second. Forget it.

So how do you define maths, please explain.
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NS
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(Original post by sdt)
Actually I thought those quantities were dimensionless for a second. Forget it.

So how do you define maths, please explain.
I win.

I like this one: short form of mathematics. :p:
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sdt
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That's too vague to use.

Anyhow, those constants are derived with algebra and a few measurements, arguably maths.

I'm just saying that a real number can just "exist" independent of our existence - which would clearly indicate maths is "discovered". Don't make me get all anthropic on your arse :rolleyes:
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Lothar Warlock
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(Original post by Oddball)
Difficult one...maths is a sort of tool, isn't it? Tools are invented and discovered.
Tooks are invented, in most cases. But yes, I strongly agree - Mathematics is a tool that was invented anddiscovered.
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dkdeath
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I would say invention, because discovery is the proccess of introduction of objects into your own reality. Invention is the procces of using existing "objects" to create something new

As for maths i see it as just concepts, concepts which sometimes are just to represent quantities (eg numbers) or to model the rate of change of these quantities, differentiation. As these concepts are created from existing objects/realities i would say its an invention.
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sonofdot
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Paul Erdos believed that God (whom he called the "Supreme Fascist") had a Book containing all the most elegant mathematical proofs, and I think that sums it up nicely. The mathematics has always been there, we just invented a way to describe it. For example, if one was to draw two dots, followed by three dots, followed by five dots, followed by seven dots, and so on, any intelligent being in the universe would be able to work out that you were talking about the primes.
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SsEe
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All maths can be traced back to set theory (so I've been told). There are a collection of axioms - the ZFC axioms of set theory. The structures that we're familiar with (groups, fields, the natural numbers) sit within a model for these axioms (a model for a collection of axioms is just a "system" in which they all hold). Eg the natural numbers can be interpreted as 0={}, 1={{}}, 2={{},{{}}}, ... and one can talk about induction and arithmetic etc. So whether mathematics is invented or discovered boils down to : Where do the ZFC axioms come from?

When I was first told the axioms of set theory, the lecturer said something like "We use these axioms because they allow us to prove things that we'd like to be true". Some of the axioms are non intuitive. Example, the axiom of foundation. Without it, we can't disprove the existence of a set having itself as an element. Humans have decided that the existence of such an object is bad so they've added this axiom. We've picked those axioms, trying to keep their number to a minimum in order to maximise the chance of consistency, yet including enough to allow us to do things we'd like to be able to do. This sounds invented. But are the axioms written into the fabric of the universe? If so then have we merely discovered a subset of these axioms that we can easily work with and derive known mathematics from?

Edit: I think maths is invented. But the starting point is things that are true in the universe. This explains why maths works so well at describing the universe.
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εїз pinga εїз
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Discovered
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El Stevo
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Back to Godel. If proving everything requires an infinite number of axioms, then do you discover or invent these axioms? If you have n axioms and know a theorem to be true, but need something else to be true, then have you discovered this something else to be true or invented it?

As Profesh said, it's a cycle of invention and discovery. All inventions are based on what you know from what you have earlier discovered.

As for Fred finding i. That's easy. Granted he was probably a couple of thousand years older at this point and uncovered the concept of negative numbers and multiplication. He saw if you multiplied a number by itself you get a positive number. He knew that both positive and negative numbers existed in a scalar fashion, so there is no real reason that multiplying something by itself wouldnt give a negative number. None of these numbers to date had that property, but that didn't mean it to be not true. He knew it was out there, so whatever it was, he defined it to be i
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DoMakeSayThink
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(Original post by El Stevo)
Back to Godel. If proving everything requires an infinite number of axioms, then do you discover or invent these axioms? If you have n axioms and know a theorem to be true, but need something else to be true, then have you discovered this something else to be true or invented it?

As Profesh said, it's a cycle of invention and discovery. All inventions are based on what you know from what you have earlier discovered.

As for Fred finding i. That's easy. Granted he was probably a couple of thousand years older at this point and uncovered the concept of negative numbers and multiplication. He saw if you multiplied a number by itself you get a positive number. He knew that both positive and negative numbers existed in a scalar fashion, so there is no real reason that multiplying something by itself wouldnt give a negative number. None of these numbers to date had that property, but that didn't mean it to be not true. He knew it was out there, so whatever it was, he defined it to be i
I think it's hard to argue that Fred didn't invent 'i', in that case, but I won't go in to that.

In your first statement, you mention an infinite number of axioms. Gödel's proof, though, doesn't explicitly say you need an infinite number of axioms. It says you need an infinte number of sets of axioms. As there can be found a Gödel sentence (or an infinite number) in any of these sets of axioms, you can construct the set of all sets of axioms from those needed to prove every Gödel sentence. Therefore there will exist sets of axioms that are infinite. (Sorry, that was more me thinking out loud than anything...) So, lets say you've got your set of axioms and you've found something you need to prove but you can't. I agree, you've discovered that thing, but all you've discovered is that your invented set of axioms is not sufficient to prove it. So beyond the axioms, I think maths is a process of discovery. As the axioms are such that all these discoveries can be related back to them, then the discoveries can boil down to invention + logic. The axioms themselves, though, I think they've got to be invented. There must be sets of axioms that don't correspond to measurable relationships in the universe, and we invented these sets of axioms after discovering our current set of axioms to be insufficient. There doesn't seem to be anywhere from which we should discover the notion of axioms.

It gets even more complicated when you start thinking about Gödel's own theorem. It was based upon the axioms of formal logic, and the only reasons they can be ascribed validity are real world results and their corresponding meaning in language. I'm just hitting my head against a brick wall, here, because I'm now running in to the problem of what truth is, and all sorts.

I think I'm going to have to concede, here. There are simply too many premises I want to rely on but can't prove, even though they may seem intuitive. It was already a foregone conclusion that this would happen, given the gravity of the question asked, it's just fun to see how far you can get in an argument.
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Adjective
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(Original post by Profesh)
Was language invented, or discovered?
I'm very reluctant to use either word to describe language - I would be more confident in saying that it was 'developed' - which is the word I've gone with (for now) to describe maths.
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Tallon
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Invention. We're the ones who decided to make up numbers and count things didn't we?
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Kolya
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(Original post by DoMakeSayThink)
When considering a logical system, I had always assumed that any result observed within that system must be a consequence of axioms. Thinking about the word "consequence", in this context, it means "logical conclusion". But if you can observe a "Gödel truth" within your mathematical system, this would suggest it was a logical conclusion of the axioms, which it can't be, as this would lead to a proof. This has given me some trouble. I think I need to better define what a "result" is.

Then again, maybe a Goedel truth isn't a "result", but merely a pattern
We know that a certain type of statement are undecidable - this was proved by Godel. To show that some other statement is undecidable, we could find a function that takes our second statement to the first statement. So we are proving "Statement 2 can be mapped to a type of Statement 1" is true; this says nothing about whether Statement 2 is a logical conclusion from the axioms or not.
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