Math: Did we discover or create it?

[Pyrotex]

Originally Posted by woog

...Perhaps we could narrow this further: the concepts commonly referred to as 1, 2, 3... did we create these concepts? What about the relationships between these objects?

Well, Socrates and Plato were pretty sure about what they (Mankind) were doing, and they lived moderately closer to the beginnings of math than we do.

These ancient Greeks concluded that there existed "ideals" that we could discover. A person could create a square or a triangle, but only because we had come to understand that there was an "ideal square" and an "ideal triangle" that had previously been discovered, or un-concealed or conceptualized by the mind of Man. These "ideals" existed from the beginning of the Cosmos, long before Man came upon the scene. There were also ideals of "beauty" and "love" etc.

To them, math was an exercise in "discovery", not an exercise of "invention" or "creation". But that was just their humble opinion, right?

[woog]

I agree totally with you.

Though now that you mention it, and with a bit more thought, I'm finding unsease with "discovered", "created", and "exists".

When I think about the meaning of "discovery", I think of "finding what is already there" or something along those lines. But when applying "already there" to mathematics in what sense do I mean? For instance in what sense does the complex number i exist, if at all?

As an interesting aside, mathematicians I know often describe research as "creating new mathematics". I get the impression that most believe what they do is discovery, even though they don't refer to it this way. For instance instead of "discovering a new theorem" its "proving a new theorem", and "creating new branches of math" instead of "discovering new branches".

I just want to add finally that coming across new ideas in mathematics I sometimes feel they seem a bit arbitrary and contrived at first, but as I delve deeper into the theory and begin to glimpse the beauty of its implications, I start to think that perhaps there is nothing arbitrary about these definitions at all. One example is group theory and perhaps even early algebra.

[Qfwfq]

Pyrotex, (sorry for getting back so late!) I'll point out that I wasn't talking bases I was talking about algebra, group theory and the likes. I tell you, mathematicians don't give a damn about reality. Really.

Definition of language? Forget what those silly dictionaries say. Here's my effort:

language: noun Any means of comunicating ideas, concepts, information, opinions, emotions, instructions, insults etc. from one individual to others that share the symbolism involved. May also be a tool for manipulating the represented asserts and deriving what conclusions may be drawn and/or for persuading others of these. Must have a lexicon with associated semantics, may have syntax and grammar.

How does that sound? How about a gesture with the hand accompanied by a throaty grunt and a grimace? There can be a lot of semantic variety there, according to subtle differences in the gestures and grunt.

Languages that we speak and write arose out of practical necessity, just like math. In comparison though, math has gone quite far beyond practical necessity. Did somebody "discover" English, Hindi or Japanese?

[Qfwfq]

Originally Posted by woog

As an interesting aside, mathematicians I know often describe research as "creating new mathematics". I get the impression that most believe what they do is discovery, even though they don't refer to it this way. For instance instead of "discovering a new theorem" its "proving a new theorem", and "creating new branches of math" instead of "discovering new branches".

I just want to add finally that coming across new ideas in mathematics I sometimes feel they seem a bit arbitrary and contrived at first, but as I delve deeper into the theory and begin to glimpse the beauty of its implications, I start to think that perhaps there is nothing arbitrary about these definitions at all. One example is group theory and perhaps even early algebra.

By the nature of math being very much based on consequentiality, which exalts its being a tool as well as a way of comunicating, there certainly is a lot of discovery in it, this is the very reason it is so much a topic of research.

Basically, once you have defined and constructed (a sine qui non by the modern point of view), there begins the work of deriving the consequences. This may make the branch more or less interesting or exciting. Given a "framework", amazing and beautiful things may be discovered, a conjecture may turn out to be true or false. These are by no means arbitrary but the framework is. What may be less arbitrary about frameworks is whether a given one will be exciting and challenging to work out... or trivial!

[kamil]

In a way we discovered math, and in a way we created it.
The Universe works mathematicaly, whether we exist or not. So in that sense the math concepts were always around. But we were the first to put these concepts on paper, and then extend their uses and meanings.
We may have created the language math. Just like we created the word 'apple'. But that doesnt mean we created an apple. But now we are able to form sentences using the word, use it when playing scrablle.

I think with maths its simimlair.

[Qfwfq]

Originally Posted by kamil

Just like we created the word 'apple'. But that doesnt mean we created an apple.

That's the idea, I distinguish between the language and what it might describe. But I differ:

Originally Posted by kamil

The Universe works mathematicaly, whether we exist or not.

This is why math is helpul in understanding reality. Strictly, I ought to say a part of math, that which describes the universe.

Originally Posted by kamil

So in that sense the math concepts were always around.
We may have created the language math.

What are "math concepts"? Is an apple a "concept"? Math is the language and method, not the apple. Numbers are handy for counting apples, this doesn't mean that numbers are apples or that apples are numbers. Neither is an RC circuit a differential equation or vice versa.

[Tormod]

Originally Posted by kamil

The Universe works mathematicaly, whether we exist or not. So in that sense the math concepts were always around.

How do you know that the universe works mathematically? We interpret it that way, but that does not mean that the universe does.

[vin]

First want to say Hi, This seems like an interesting forum.
My take on this matter. I've thought about it for a few years now as a couple of classmates and I used to have this discussion over beers occasionally.

First I would start by defining math,

Math: mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions

The definitions and axioms themselves can be either discovered or created. I would say most are created by the mathematician with inspiration from some outside source, be it the bahavior of numbers, the physical world, or some other system that the mathematician is studying.

The deductive reasoning is discovered. A mathematical system is deterministic, the system itself behaves independently of how it is observed.
If a theorem is proven and the mathematician dies before writing the proof does it make the theorem any less true?

[Pyrotex]

Originally Posted by Jay-qu

yeah i would agree that it is discovered but there are some parts about math that have only come about directly because of us humans. for example we use base 10 as our default system - I assume because we have 10 fingers and 10 toes...

Actually, only a miniscule part of math depends on base 10 or any particular base. That tiny part is merely the writing of numbers, what symbols are used to express the quantity 12 for example.
12(base10)= "12"
12(base12)= "10"
12(base8)= "14"
12(base3)= "110"
12(base2)= "1100"

That tiny part also includes what written symbols we use. Is the symbol for "one" written as "1", "T", "(", "a", "-", or whatever?

The rest of math: arithmetic, algrebra, trig, calculus, matrices, ... ,
AND all the science that uses math: physics, chemistry, statistics, ... ,
are all independent of how you write a "one" or what base you use.

[CraigD]

Originally Posted by vin

…Math: mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions … A mathematical system is deterministic, the system itself behaves independently of how it is observed.
If a theorem is proven and the mathematician dies before writing the proof does it make the theorem any less true?

Welcome, Vin! As you might gather from my post count, I too find this an interesting place to hang out and discuss all sorts of Math/Science related stuff.

I agree with your definition of Math. I prefer the term “formal system” to “mathematical system”, but believe we’re both referring to the same thing. I take the additional step of asserting that reasoning is itself a collection of axioms in a formal system that, in the sense that it is used to create the description of every formal system ever described by a human being, has existed effectively forever. Whether this “bootstrap” formal system was discovered, created, and if so by what or whom, is a question that gets very deep very quickly!

PS: Try posting to the introductions forum, if you’d like to, well, … introduce yourself.