Maths - invented or discovered? Watch

El Stevo
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#41
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#41
(Original post by DoMakeSayThink)
That is, working with our current set of logical axioms there are results that are certainly true but impossible to prove.
From my very limited Godel - I touched on it 3 minutes ago, said results are also impossible to disprove. If you know something, but don't know why, which essentially not being able to prove or disprove it means, can you claim to have invented this or have you merely discovered it? Surely inventing something implies a deliberate construction?
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DoMakeSayThink
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#42
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(Original post by El Stevo)
From my very limited Godel - I touched on it 3 minutes ago, said results are also impossible to disprove. If you know something, but don't know why, which essentially not being able to prove or disprove it means, can you claim to have invented this or have you merely discovered it? Surely inventing something implies a deliberate construction?
Well yes, if something is true it follows that you can't disprove it.

Thinking about this is proving a little troublesome, and I'm not sure how far I'll get before giving in to sleep - I have got work tomorrow.

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.

Oh my.... I'm drowning in a sea of not knowing what the hell I'm talking about, and having way too many ideas knocking about my head. I'm gunna give up for now, and maybe have a think/read tomorrow.
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Tut.exe
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#43
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maths is not man made -.-' the language to communicate with it is.
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DoMakeSayThink
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(Original post by Tut.exe)
maths is not man made -.-' the language to communicate with it is.
While that's a nice theory, do you have any arguments to support it?
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Robob
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#45
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I think maths was invented, since it is essentially a human construct although very useful for describing the world.
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Tut.exe
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#46
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(Original post by DoMakeSayThink)
While that's a nice theory, do you have any arguments to support it?
because shapes, numbers, and relationships exists even if we dont have a language for it. before we spoke one, there was already one. before the golden ratio was found, it was already widely evident in nature. and the golden ratio for example wasn't created... it was found. so was the size of the earth.

Edit: Im trying to say mathematics is a property of the universe found by man... not a mere description made by man.
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Robob
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(Original post by Tut.exe)
because shapes, numbers, and relationships exists even if we dont have a language to it. before we spoke one, there was already one. before the golden ratio was found, it was already widely evident in nature.
They don't exist until you define them...

Maths is basically a logical progression from some basic axioms. Those axioms are defined, they don't 'exist'. So surely maths was invented not discovered, it's not there until you define what numbers are etc.
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Robob
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(Original post by Tut.exe)
Edit: Im trying to say mathematics is a property of the universe found by man... not a mere description made by man.
Got any evidence for that?

Maths is used to describe the universe, nothing more.
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Tut.exe
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(Original post by Robob)
They don't exist until you define them...

Maths is basically a logical progression from some basic axioms. Those axioms are defined, they don't 'exist'. So surely maths was invented not discovered, it's not there until you define what numbers are etc.
as i've said, it's a property of the universe. and it's not just logical progressions from some basic axioms. If it was, it wouldve been so easy to define the whole of it and scientists wouldnt use it to describe or study stars or use statistics to understand nature if it was just a logical progression from some basic axioms. Mathematics is best at describing the universe because the universe is runned by it dont you think?

Just like a door, it has properties and we describe its properties using the language english but it doesnt lose its properties, neither can we say it doesnt exist if we dont hold a language for it, or for describing it.
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sdt
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(Original post by El Stevo)
axioms not related to the real world?

Ok, I'm going to introduce you to Fred, the original mathematician. He needed to make a fire, but was crippled so couldn't fetch the wood, so he get's Barney. He says he wants to make a fire. Barney asks 'how big?' At this point they are stumped (pardon the pun).

Fred takes a stick and places it down. He says "Right, Barney, this is one stick."
Barney says "ok"
Fred removes the stick and says "This is the absence of one stick, also known as zero sticks"
Barney says "ok"
Fred finds another stick and puts it separate from the first stick, and asks Barney "What's this?"
Barney says "One stick"
Fred puts them next to each other and asks "What are these?"
Barney says "That's one stick and one stick"
Fred says "Aha. But for short, let's call this two sticks, and let's define putting two lots of one stick together as addition"

Ad infinitum, and we have a counting system based on real world axioms.
When Fred can define i using sticks, I will eat my shoes.

I'll go with discovered. The notion of a wavefunction is more than enough evidence.
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Robob
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(Original post by Tut.exe)
as i've said, it's a property of the universe. and it's not just logical progressions from some basic axioms. If it was, it wouldve been so easy to define the whole of it and scientists wouldnt use it to describe or study stars or use statistics to understand nature if it was just a logical progression from some basic axioms. Mathematics is best at describing the universe because the universe is runned by it dont you think?
Sorry what? Give me one reason why it's a property of the universe?

Scientists use it because you can model the universe using mathematics, well at least to an extent. That doesn't mean that the universe is 'runned' by mathematics.

Your second sentence makes no sense, please elaborate
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Profesh
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(Original post by DoMakeSayThink)
I'm finding it quite difficult to come up with arguments to support the notion of language being discovered. Can you give any examples?
I tend to regard mathematics as a language: it has syntax; it has definitions; its has terminologies; it describes relationships which may hold true in the corporeal realm (and so inform physicists), but which can also be abstract. 'Discovery' in mathematics occurs when mathematical rules/processes/algorithms are applied to values (which we also invent, though they - and the processes - may well correspond with something tangible), and this can evince new axioms; and so there is a 'science of maths', if you will, comprising various axioms which can themselves then be applied in novel configurations, and by which we are able to take ever more for granted. It's a cycle of 'invention' and 'discovery' operating in tandem.

Though I'm not a mathematician, mind, so this interpretation may be grossly inaccurate.
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Robob
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(Original post by sdt)
When Fred can define i using sticks, I will eat my shoes.

I'll go with discovered. The notion of a wavefunction is more than enough evidence.
The notion of a wavefunction is concept used to describe the motion of an electron (for example). There are other ways, for instance you can use the 'sum of all paths' to describe its motion, whereby the electron travels along every possible path all around the universe, and you sum the result.

But just because something can be described by mathematics doesn't mean that mathematics exists outside of human concepts.
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Robob
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(Original post by Profesh)
I tend to regard mathematics as a language: it has syntax; it has definitions; its has terminologies; it describes relationships which may hold true in the corporeal realm (and so inform physicists), but which can also be abstract. 'Discovery' in mathematics occurs when mathematical rules/processes/algorithms are applied to values (which we also invent, though they - and the processes - may well correspond with something tangible), and this can evince new axioms; and so there is a 'science of maths', if you will, comprising various axioms which can themselves then be applied in novel configurations, and by which we are able to take ever more for granted. It's a cycle of 'invention' and 'discovery' operating in tandem.

Though I'm not a mathematician, mind, so this interpretation may be grossly inaccurate.
I'd broadly agree with that.
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sdt
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(Original post by Robob)
The notion of a wavefunction is concept used to describe the motion of an electron (for example). There are other ways, for instance you can use the 'sum of all paths' to describe its motion, whereby the electron travels along every possible path all around the universe, and you sum the result.

But just because something can be described by mathematics doesn't mean that mathematics exists outside of human concepts.
That is the whole point though. The measured value (well, the probablility anyway) of the wavefunction comes out with a very snug fit to the mathematical model.

How about the dimensionless parameters of the universe? I'm pretty damn sure they existed long before any caveman tried to divide 1 into 137 parts.

Either way, I can't see how an admissions tutor would honestly give a flying ****.
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Tut.exe
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(Original post by Robob)
Sorry what? Give me one reason why it's a property of the universe?

Scientists use it because you can model the universe using mathematics, well at least to an extent. That doesn't mean that the universe is 'runned' by mathematics.

Your second sentence makes no sense, please elaborate
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.
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generalebriety
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#57
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(Original post by sdt)
How about the dimensionless parameters of the universe? I'm pretty damn sure they existed long before any caveman tried to divide 1 into 137 parts.
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...
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Tut.exe
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i just believe it's discovered by human beings, being the language of the universe.
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Robob
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(Original post by sdt)
That is the whole point though. The measured value (well, the probablility anyway) of the wavefunction comes out with a very snug fit to the mathematical model.

How about the dimensionless parameters of the universe? I'm pretty damn sure they existed long before any caveman tried to divide 1 into 137 parts.

Either way, I can't see how an admissions tutor would honestly give a flying ****.
Well yes, the model is good... what's your point?

Well they only exist within the mathematical framework.

And I agree with your last point.
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Kolya
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(Original post by Swayum)
This is what separates Maths from science. Science is about discovery - you can't make science. You can make Maths though - it's just a bunch of definitions. Maths is not a science at all, it is, if anything, a language (the language of science).
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:
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