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Maths - invented or discovered?

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benwellsday

1 + 1 will always be 2

Depends what you mean by +. Consider the example of adding being the process by which two objects are put together. This could be 1 (adding lumps of clay) 2 (putting two bananas together) or 3,4,5... (depending on the fertility of the objects in question)

Reply 21

Svefngenglar

In my view, maths is a human invention inspired by nature. Processes such as counting are artificial, since the setting of a unit to the number one is in itself artificial. Numbers, strictly speaking, only exist in an abstract state. Similarly, any geometric application to real life is undermined by the fact that structures such as a perfect straight lines, perfect right angles and perfect circles do not exist. Besides, there are entire fields of mathematics with no clear practical application, such as complex number algebra.

But then again, whatever you mention in your personal statement, the important thing is defending your view convincingly and interestingly. There's no 100% right answer, though I would say the case for maths being invented is stronger than the alternative.

Reply 22

username167718

Mathematical methods were discovered. Mathematical notation was invented.

I say this because everything in maths (besides the notation) has always existed, long before humans were around. 1+1 has never equalled 3, but no one had ever written 1+1=2 until someone invented a way of writing it.

Reply 23

Swayum

Maths is definitely invented as far as I'm concerned. You can't "discover" a number 3, can you? We've defined the concept of a "3" (which can be REPRESENTED by, say, 3 apples - but that's still not a "3"). Because we define addition, multiplication etc Pythagoras' theorem holds, but as soon as you change that definition, the theorem would not hold. So then how can you say we have "discovered" Pythagoras' theorem if its entire nature changes as we change our definition?

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).

Reply 24

Charlybob

Swayum

That makes no sense. At all.

How could you say Pythagora's theorem wasn't discovered? It's unchanging. The square of the largest side is always going to be the same as the sum of the other squares, whatever we say the numbers mean. You can reword it, but it's still a fundamental that isn't going to change at all.

3 still existed before we named it. Dogs can notice if something's missing. Give them 3 balls, take one while they're gone, they'll know. They don't know what a 3 is, but they know a ball is gone, because a 3 exists beyond the definition we give it.

Rewording something in maths doesn't mean that it's not going to hold afterwards. All it means is we use different words to describe it.

Reply 25

generalebriety

Charlybob

That makes no sense. At all.

How could you say Pythagora's theorem wasn't discovered? It's unchanging. The square of the largest side is always going to be the same as the sum of the other squares, whatever we say the numbers mean. You can reword it, but it's still a fundamental that isn't going to change at all.
What's a triangle? I've never seen one. Nor have I ever seen a right angle. I've certainly never seen a right-angled triangle, drawn squares off each of its sides, measured their areas and shown that the two smaller ones add up to the bigger one, it's completely impossible.

Charlybob

3 still existed before we named it. Dogs can notice if something's missing. Give them 3 balls, take one while they're gone, they'll know. They don't know what a 3 is, but they know a ball is gone, because a 3 exists beyond the definition we give it.
That's not a 3. That's the concept of a bijection through time. Scatter a load of balls over the floor, take a photo, then remove one, and take another photo. Hold the two photos side by side and you can see one has been removed, even if you don't know how many balls there are.

SimonM

Or, of course, we could be taking the (now rather outdated) set-theoretical meaning of +, where A + A = A for all A (which is now known as a union).

Reply 26

JamieP

Swayum

Maths is not a science at all, it is, if anything, a language (the language of science).

I'd agree with that. Maths is a way of looking at things, the underlying relationships were always there but we created a universally standard way of describing and communicating them.

As such I'd say that the concepts that we use maths to describe (eg pi and its relationships) already existed and were discovered by humans, but that the notation we use to describe them was invented/created by humans

Reply 27

Charlybob

generalebriety

What's a triangle? I've never seen one. Nor have I ever seen a right angle. I've certainly never seen a right-angled triangle, drawn squares off each of its sides, measured their areas and shown that the two smaller ones add up to the bigger one, it's completely impossible.

Uh, no it's not?

I honestly haven't got a clue what you're trying to say here. Like I said with the 3, the shape we call a triangle can still exist without being defined as such, and so does the angle. What you've said amounts to "pythagoras doesn't work if I don't define a right angled triangle as such". But it still does, so you want to try clarify that a little?

generalebriety

That's not a 3. That's the concept of a bijection through time. Scatter a load of balls over the floor, take a photo, then remove one, and take another photo. Hold the two photos side by side and you can see one has been removed, even if you don't know how many balls there are.

This one may make me sound like a complete idiot, I had to look bijection up on wikipedia so if I go off on a tangent which has nothing to do with what you're talking about because of taking the wrong definition, sorry. (Please don't bother with "it's not a tangent if I change the definition of a tangent" comments. I will lose the will to live if anyone does.)

But again, the bijection exists beyond our definition of it. One (ten, fourty-seven, however many you want to define the single as ¬_¬) has gone (or been added if you choose to use that definition of a reduction (or increase if you choose that definition of the definition)). You're not changing maths by re-defining it. The fundamental system still works in the exact same way.

All we change by altering our definitions, is the words we have to use if we want to describe the system to someone else.

Reply 28

manderlay in flames

Different cultures come to (exactly) the same conclusions... discovered.

I'm loving the argument from a dog's perspective of number, really original

Reply 29

generalebriety

Charlybob

This one may make me sound like a complete idiot, I had to look bijection up on wikipedia so if I go off on a tangent which has nothing to do with what you're talking about because of taking the wrong definition, sorry. (Please don't bother with "it's not a tangent if I change the definition of a tangent" comments. I will lose the will to live if anyone does.)
Tangent? What are you on about?

A bijection is a one-to-one relationship. You try to match up each ball in the first photo with exactly one ball in the second photo, and realise that however you do it, there's a ball left over in the first photo... which means there must have been one more ball in total in the first photo. This holds true whether it's 3, 50 or 7,924 balls, and we don't need to count them to find out. In fact, in practice, this is exactly what you'd do. You wouldn't count the first photo to find out there's 37 and the second to find out there's 36, you'd just look for a missing ball.

Reply 30

El Stevo

Maths is like the ocean - it's clearly there to be explored and you don't need anything to cover the surface. However, to get really deep into it you need to build a submarine and dive down. For earlier mathematicians this submarine may have been the concept of counting, which lead to addition, and ordering. Anything you 'invent' to probe maths further is based on things you have already discovered. You can only build a proper submarine when you know it needs to be waterproof, hold pressure and be able to vary it's mass. You only know this because of your experience of the ocean.

And thats a 2.30 am anaology for you. My PS included my belief that maths is the language of science. It also likened it to music.

Reply 31

generalebriety

Charlybob

This one may make me sound like a complete idiot, I had to look bijection up on wikipedia

...also, while I hate to be patronising, if you don't know what a bijection is, you haven't done a basic first-year university maths course, which probably means you have no experience of pure maths. Groups, rings, fields, spaces, metrics, topologies, that sort of stuff. I find it very difficult to believe that the concept of a metric space (look it up, it's not difficult), for example, was discovered. It's a lot more believable with applied maths (though I am playing devil's advocate here to attempt to show you it's not quite as simple as you're making out), but with a lot of pure maths you simply have to stand back and say "actually, that gives some nice results, but there's no way someone just happened to fall across that; that's the result of some crazy mathematician somewhere making up interesting crap".

Reply 32

DoMakeSayThink

generalebriety

Oh, god. This is a huge philosophical question that's been going on for thousands of years, and you want the f38ers to answer it? :p:

I say - put very simply - definitions are inventions, theorems are discoveries about those inventions. But of course that's a hideous oversimplification. Some definitions are made with theorems in mind, which makes the definition more of a stepping stone to 'inventing' the theorem (i.e. inventing a place where the theorem can live happily); some definitions certainly are discovered, e.g. differentiation was 'happening' in, say, velocity and acceleration, long before we discovered a way of notating it and working with it. And a lot of it, I would guess, was just pure chance.

Reply 33

Ewan

I think its a system we created to explain how the world works. I think thats why it works, you see its not like Science. Science can be wrong, maths can't, because its something we created. It can't be wrong, because we make up the rules, with science, nature makes up the rules, and we have no control over nature!

Discoveries are in my opinion physical things which are found. An atom is a discovery, finding out that the world isn't flat is a discovery, deciding you have two apples in your hand is a way of representation. Deciding that cutting an apple in the centre creates two halfs is a way of explaining what you've done.

A discovery is something thats found, an invention is something thats created.

tomthecool

Mathematical methods were discovered. Mathematical notation was created.

Spot on.

Reply 34

El Stevo

DoMakeSayThink

I'm really inclined to lean towards invention, because all maths is based upon axioms which do not all relate to the real world. In addition, isn't it possible to create new systems of mathematics from different axioms? This would suggest that the axioms employed aren't discovered, as otherwise alternatives wouldn't be available.
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.

Reply 35

Crystaltears

Erm...good question!!
Well on my last ever AS maths lesson my teacher decided to teach us some degree level maths and he did all this wierd stuff with i and pi and the answer came to 1. So he was saying that maths was discovered because it was so strange that all these complicated numbers would divide to equal 1.
But then again, numbers aren't real, so it's invented because it's something we have made up.

Maybe numbers are invented, but the concepts were discovered? (oh, that's like the opposite of what you said haha!)

Ignore me, i'm crap at maths!

Reply 36

DoMakeSayThink

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.

Sorry, my first post is incorrectly worded. It should read "all maths could be based upon axioms which do not relate to the real world". To be perfectly honest, I'm quite out of my depth here, but from the very little I've heard of Gödel's incompleteness theorem, I understand that it's impossible to prove every true result without infinitely many sets of axioms. That is, working with our current set of logical axioms there are results that are certainly true but impossible to prove. Further, to prove all true statements, one would need an infinite number of sets of axioms. Please correct me if I'm wrong about this. Taking my interpretation, it is clear that one of the infinite sets of axioms would not be related/based upon the real world.

Reply 37

Valkyrja

I think the most beautiful thing about maths is that it is infinite, at the lower levels, you can of course understand it, but what about infinity? Maths is in a lot of ways quite a creative subject. I wish I was much better at it than I am.

Like that whole thing of 1 + 1/2 + 1/4 + 1/8 + 1/16 etc. in infinity = 2. How poetic is that? It's quite lovely really.

Reply 38

Profesh

Adje

I'm writing my personal statement, and I'm trying to say something about maths, and why it's good.

But I'm having a hard time ascertaining whether the concept/s of maths is human invention or human discovery. I have a feeling that the deeper you delve into mathematics (i.e. as you go further into abstraction), maths becomes more invention than discovery - but at its simplest - counting, say - it is more discovery than invention, since animals other than humans have at least some idea of differences in quantity.

Your thoughts?

Was language invented, or discovered?