Have we discovered or invented maths?

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In summary, there is a debate about whether mathematics is an invention of humans or a discovery connected to nature. Some believe that we have created math to suit our needs and it is a language we use to describe the natural world, while others argue that mathematics has always existed and we have just invented a language to express it. It is also suggested that math is both a discovery and an invention, and that it is a way for us to understand and decode the principles of nature.
  • #1
C0nfused
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Hi everybody,
This must have been discussed before but I would like to hear your opinions about the above question. Are mathematics just an invention, a creation of humans that helps them in their everyday life, or are they actually connected to nature, and are part of it that we just happened to discover? I personally think that we have created maths, and certain needs made humans do so, and during the years we have changed and defined them in such a way, that we can apply them in many different aspects of nature and our lives. And as for theoritical mathematics, theorems and properties are,in my opinion, just consequences of the definitions we have created, some of which have been found and proved ,while some others not.
What do you think?

Thanks
 
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  • #2
Invented or Discovered.

Hopefully it does not deviate us from reallity. I mean if maths is wrong then its like a piraymid; One wrong block at the bottom and the entire structures looks stranger than facts.
 
  • #3
C0nfused said:
... a creation of humans that helps them in their everyday life, or are they actually connected to nature, and are part of it that we just happened to discover? ...

There have been similar threads here. I always throw into the mix the idea that I will expect any space faring race of extraterrestrials that we may run into "out there" to have a plane geometry (for example) that will have a great deal of overlap with our own plane geometry. If they can design and build spacecraft , they will most definitely not claim that for a planar right triangle residing in a flat space, the cube of the hypotenuse is equal to the sum of the cubes of the sides.

So at least to a degree, I believe that the math we come up with is indeed connected to nature. Remember that the very word "geometry" means "measuring the earth."
 
  • #4
Relationships between things exist in the universe. Selecting and relating them to each other is the work of minds. But minds can also invent relationships that are not between outer things, but between thoughts. So the answer is both: people may discover whether the geometry of space is euclidean or not, but they may also discover there is a principle bundle over spacetime with such-and-such a group, not something the intrepid cosmo/astro-naut will ever encounter.
 
  • #5
selfAdjoint said it very nicely: mathematics is about the relationships in the abstract- what are all the possible ways that things can be related to one another.

As to whether we invent or discover things in mathematics, the answer is BOTH!
Mathematical theorems are "invented" when we first construct the "mathematical structure": develop the axioms, undefined terms, etc., then "discovered" when we prove them. Yes, that requires that we invent things we don't know about! The term "emergent properties" is very evocative here: properties that were inherent in the structure but that we did not know about.
 
  • #6
We invent Mathematics.

Numbers don't have a separate existence from the mind: The Earth and sun were here long before us but the equations which describe the orbit of the Earth were not.

Von Neumann considers the natural numbers as a play on empty sets by the mind: The mind ponders the empty set. Conceiving the empty set generates the number 1. Conceiving the set containing the empty set and the empty set generates the number 2 and so on. Thus the natural numbers from his perspective are created independently from any physical existence.

Mathematics fits nature so well because the geometry of Mathematics itself and that of nature are very similar: The real numbers system is dense: Between any two real numbers lies another. Nature appears dense as well: no smallest small nor largest large. Nature is "nested" (plans within plans). So too is math: Think of the chain rule. Nature is non-linear, so too is the geometry of math: Make one small mistake in solving a problem and it's not just a small error in the results.

It makes perfect Darwinian sense for a human mind to conceive of a math which fits nature from the perspective of survival: When in New York, act like a New Yorker. When the human brain finds itself in a non-linear world, a successful survival strategy it seems to me would be to devise a system of metaphors (mathematics) which fit that non-linearity.
 
  • #7
C0nfused said:
Hi everybody,
This must have been discussed before but I would like to hear your opinions about the above question. Are mathematics just an invention, a creation of humans that helps them in their everyday life, or are they actually connected to nature, and are part of it that we just happened to discover? I personally think that we have created maths, and certain needs made humans do so, and during the years we have changed and defined them in such a way, that we can apply them in many different aspects of nature and our lives. And as for theoritical mathematics, theorems and properties are,in my opinion, just consequences of the definitions we have created, some of which have been found and proved ,while some others not.
What do you think?

Thanks
Math is both a discovery and an invention. We invented a universally accepted code to express the discovery of mathematical principles in terms others will understand. Math is the language we use to encode those principles.
 
  • #8
Thanks for all your answers.
 
  • #9
Math has always existed, we have just invented the language in order to speak it.
 
  • #10
Humans invented math, just as humans invented god, or just as humans invented internet.

Mathematics is the way we use to express the languange of nature. It is not, I repeat, it is not the languange of nature, it is the way we describe it.
 
  • #11
Book of numbers

C0nfused said:
Thanks for all your answers.


The astrophysicist Paul Davies asked this same question in his book, "About Time". It is an question that I cannot answer for certain but I am in the camp of those who believe that mathematics is an invention of humans.
 
  • #12
I am trying to learn the math of modern physics by reading Penrose's Road to Reality, and the author is clearly a Platonist.

Maybe he'll have me converted by the time I'm finished with the book, but right now I'm incredulous, and believe that math is a construction. If math existed in a timeless and an independent manner, how come we don't have timeless and independent meanings for our symbology? Or to rephrase it, why is not easier for us to have timeless and independent meanings in our symbology?
 
  • #13
Is it possible that Math will fall short to answer all questions ?

If we have invented Maths then will there be any other invention that will be more significant ?
 
  • #14
Telos said:
I am trying to learn the math of modern physics by reading Penrose's Road to Reality, and the author is clearly a Platonist.

Maybe he'll have me converted by the time I'm finished with the book, but right now I'm incredulous, and believe that math is a construction. If math existed in a timeless and an independent manner, how come we don't have timeless and independent meanings for our symbology? Or to rephrase it, why is not easier for us to have timeless and independent meanings in our symbology?

I think you miss the point: the objects exist in this platonic realm, but the symbols are merely ways of referring to the objects, the language of referral is not itself part of the platonic realm. Langauge is a human invention, so it will change, the things it refers to may not.

As it happens I do not adhere to the platonic theory. But, as someone else has observed, it doesn't matter whether or not you are a platonist or a formalist, the maths is still the same. However, as the book is about physics, platonism is not surprising.
 
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  • #15
matt, Penrose does make it clear that "Platonism" is really just a way of agreeing to the "objectiveness" of mathematical objects.

I know that I am missing the point. lol, it's just that I feel I understand and can manipulate mathematical objects quite well, but the language gets in my way. I sort of just feel the objects interact with one another and in my mind outputs a solution. I have heard that autistic savants do something similar - but of course I'm no where near that developed.

So, I really should be a Platonist because I experience math as independent from its language, but then because I haven't accepted the way much of math is written, I've effectively closed myself from expressing it. And, if I can't express it - do I know it? No, not according to pedagogy.

So, is it possible for one to know math but at the same time not know how to express it?

It seems that if you answer "yes," you're a Platonist, and if you answer, "no," you're a formalist.

But, as someone else has observed, it doesn't matter whether or not you are a platonist or a formalist, the maths is still the same.

Maybe it matters for pedagogical reasons? Which are also related to communicative reasons?
 
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  • #16
No, it doesn't matter whether or not you think that mathematical objects have an independent existence in some other reality (and "in your head" isn't what that means). All that matters is, for example, that you understand that a 2-sphere has vanishing fundamental group. It doesn't matter whether you think that there is in some other universe an object that *is* a 2-sphere, and this comes down to understanding the definitions and being able to express yourself clearly. Many people con themselves into thinking they "understand" mathematics but can't explain it. No, the standard is that unless you can explain it you can't say you understand it. Some people have better intuition than others and see the results faster, and don't need to verify details since they can see an argument as to why its true. Witten for instance is widely credited with having fanatastic intuition and little time to verify his conjectures personally - they predomintantly are true, it appears.

The idiot savants you are talking about aren't mathematicians, they are arithmeticists - knowing how to take square roots in your head isn't mathematics.

I think yo'ud have to explain what you meant by "objects" and "seeing things in your head". To go back to the sphere example, there is abig difference between intuitively understanding something about the sphere by visualizing it and it actually having a platonic existence.
 
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  • #17
Thank you once again, matt. I think my relating Platonic existence with an experience "in my head" may have something to do with my forays into lucid dream studies, and the new age perception that many of these so-called lucid dreams actually take place in separate realities (or as they term it "wider reality"). I don't completely subscribe to those notions, but I can't ignore the experiences I've had that cause me to be curious of them.

You've helped me see it differently, and therefore have helped opened doors to mathematics for me. I'll repay you by understanding as much math as I can.

It makes you wonder how many Ed Wittens are out there who don't learn math because they think too intuitively, doesn't it?
 
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  • #18
C0nfused:
Are mathematics just an invention, a creation of humans that helps them in their everyday life, or are they actually connected to nature, and are part of it that we just happened to discover?

We discover the consequences of our inventions.

That logic/mathematics can be applied to the world is a result of the generality of logic, i.e. it applies to all existent things.

There are no existent things that are excluded from logic.
 
  • #19
mapper said:
Math has always existed, we have just invented the language in order to speak it.

I don't agree.

Logic and mathematics are mental phenomena, i.e. there is no logic or mathematics without mind. Indeed, there are no languages either.

There cannot be any timeless things!

Concepts are dependent on mind.
Mind is dependent on brain.
Brain is dependent on physical things.
Physical things are dependent on time.
therefore,
Concepts are dependent on time!
 
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  • #20
Read Steven Pinker's book, Language, and discover that there are strong reasons to believe that we are wired for language, hence mathematics as well. But this does not indicate there's no invention -- calculus, Fourier Series, Bessel Functions, Banach Spaces and on and on were certainly invented. It's just that we have a genetic propensity to be able to invent such things.
Regards,
Reilly Atkinson
 
  • #21
But could they have been invented differently ?
 
  • #22
Tournesol said:
But could they have been invented differently ?

***************
How so?
 
  • #23
Yes, they could (be invented differently), and this is the reason why many people think platonism is wrong. For instance R=C is completely indistinguishable from R[j] where j is any other symbol that squares to give -1 (such as -i). Almost everything in mathematics is constructed, or depends on things that are constructed to meet specific needs, and that includes the real numbers.
 
  • #24
I was trying to hint that mathematics is not really invented because in fact it could not
have been different. (And not discovered either -- there is no reasont to think those
two metaphors are the only options.

Matt: does it make sense to say that, e.g real numbers could be constructed differently and still be (in the standard) sense real numbers ? Of course we could use different symbols...
 
  • #25
There are many ways to construct the real numbers. The key result in this is the following statement(s)

F is a field if it satisfies some axioms (I won't bother saying what they are)
F is an ordered field if it possesses an ordering < (we may want a total ordering, i can't remember).
F is a complete ordered field if it satisfies another rule.

Theorem: any two complete ordered fields are isomorphic as ordered fields, and the isomorphism is unique.

We call any (totally) ordered complete field R, the real numbers.

Three models of them are:
1. the equivalence classes of all convergent sequences of cauchy sequences of rationals.

2. the set of all dedekind cuts of R

3. the set of all expressions of the form

N.d_1 d_2 d_3...

expansions in some base b, N an integer, d_i in {0,1,..,b-1} with the declaration that repeating strings of b-1's in the epxansion are not allowed (identify it with a terminating string instead)

We extend all operations in the obvious fashion.

The point of these definitions is to create a solid mathematical framework in which to do analysis. If we didn't identify terminating strings with those recurring b-1's then we would have a model of a system where the archmidean axiom would be inconsistent.

So mathematical objects are merely things that satisfy some rules. At least that is the modern interpretation of them. Some still argue that there are actually things that satsify these rules in the platonic sense, or that we are merely stating what these things in some other universe "are". To be honest it doesn't matter to mathematics directly. However, if you want a system which has fractions, and in which we can sensibly talk about sequences that have all the right properties then you'll come back to the reals, or a model of them. It is, if you like, universal with respect to this property. This is a common idea in category theory (see below).

Here's one for you to ponder: the rings R[e] and R[pi] are indistinguishable algebriacally.

A modern view, and one perhaps pushed by category theorists, is that it isn't not beneficial to talk of things being equal, only isomorphic, or equivalent - different symbols in different contexts that behave the same.
 
  • #26
Mathematics is the study of patterns, is it not? Patterns are discovered, not invented, in my opinion.
 
  • #27
I don't think mathematics is necessarily the study of patterns, whatever that may mean (it seems vague and open and guaranteed to be true given anyone's personal interpretation of it, in fact I think everything is the study of some patterns in some sense if we think hard enough), and even if it were a reasonable thing to say, it doesn't mean that the place where we're looking for these patterns wasn't invented, and thus that the pattern is invented. As I think Hallsofivy, or perhaps Hurkyl said, "we invent mathematical objects to suit our needs and then discover things about them" (apologies if that's isn't verbatim. I believe it is in a post in this thread earlier).
 
  • #28
Let's say we place mathematics between philosophy and physics. The former is certainly invented, and the latter, discovered. Why can't mathematics be a composite of invention and discovery?
 
  • #29
I think the premise of this thread has no point. Whether math is discovered or invented, you're still going to use it. That's because the formulas and postulations can be stacked up against arbitrary amounts of data and still remain consistent in their conclusions. Thus they have become axiomatic and factual in nature.
 
  • #30
matt grime said:
So mathematical objects are merely things that satisfy some rules.

Quite. And there are different ways of arriving at things (in this case reals)
that follow the same rules. But because they do folllow the same rules
we say they are all different construction of the same objects. The point being
that we are not free to construct 'new' reals that follow different rules form the old ones -- the new objects just wouldn't be reals. Hence maths is not
'created' in an artistic sense (which does not mean it is discovered in a Platonic sense; discovery vs invention is not a genuine dichotomy).
 
  • #31
Tournesol said:
Quite. And there are different ways of arriving at things (in this case reals)
that follow the same rules. But because they do folllow the same rules
we say they are all different construction of the same objects. The point being
that we are not free to construct 'new' reals that follow different rules form the old ones -- the new objects just wouldn't be reals. Hence maths is not
'created' in an artistic sense (which does not mean it is discovered in a Platonic sense; discovery vs invention is not a genuine dichotomy).
Would you agree that, say, poems are "created" in an artistic sense? Many poems, though constructed of different words and having different structures, express the same meaning, so they are semantically the same and follow the same semantical rules.
 
  • #32
Tournesol said:
Quite. And there are different ways of arriving at things (in this case reals)

that follow the same rules. But because they do folllow the same rules
we say they are all different construction of the same objects. The point being
that we are not free to construct 'new' reals that follow different rules form the old ones -- the new objects just wouldn't be reals. Hence maths is not
'created' in an artistic sense (which does not mean it is discovered in a Platonic sense; discovery vs invention is not a genuine dichotomy).


But the reason why we are "not free to construct new reals that follow different rules" is because we have made the rules that they satisfy such that there is only one complete totally ordered field.


If we just take the axioms of a field, there are infinitely many non-isomorphic examples of them.

It is a convention that once we have fixed the rules we don't change them as this avoids confusion, but if you look at cutting edge research then you'll see many different definitions with the same name competing to see which one is the one we ought to adopt.

And remember we chose the rules that the objects satisfy. Just like we can choose to write poems in tetrameter or pentameter, or maybe make it a haiku.
 
  • #33
Hi everybody,
I had some time to check this thread and I am surprised from the number of answers. By reading all of them, I first of all saw that there are many different interpretations of mathematics. I come to think that each person has a unique "understanding" of mathematics, and has interpreted math concepts in such a way that he can be more efficient in studying and solving math problems. I also agree with this:
Owen Holden said:
We discover the consequences of our inventions.
One more thought: there can't be right or wrong in this issue as there wasn't one person that "created" or "invented" maths so there is not any specific and "formal"-strict definition of it. However i don't agree with this
erraticimpulse said:
I think the premise of this thread has no point. Whether math is discovered or invented, you're still going to use it. That's because the formulas and postulations can be stacked up against arbitrary amounts of data and still remain consistent in their conclusions. Thus they have become axiomatic and factual in nature.
Answer:
I think that trying to understand the nature of mathematics will help to make you understand maths better. And why do you think that maths are definitely going to be used for ever(that's what you imply by saying that "you're still going to use it")? I know that maths has proved it's importance all these thousands of years but this doen't mean that it will stay there for ever. Or does it? (I have just remembered something i think Hardy(is this written this way?) said, that Archimides will be remembered when Aeschylus will have been forgotten,because languages "die" while mathematical truths are eternal ( or something like that!))
 
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  • #34
Tournesol said:
Matt: does it make sense to say that, e.g real numbers could be constructed differently and still be (in the standard) sense real numbers?
That depends on what is meant by the phrase "and still be (in the standard ) sense real numbers? If the term "real numbers" is no more than a tag for entities which obey the set of rules divined to be "the rules of real numbers", what does the term differently mean?

I think you are just stirring that pot of vague terms with the hope that, if you stir it enough, they will stop being vague. :yuck:

Have fun -- Dick
 
  • #35
Doctordick said:
That depends on what is meant by the phrase "and still be (in the standard ) sense real numbers? If the term "real numbers" is no more than a tag for entities which obey the set of rules divined to be "the rules of real numbers", what does the term differently mean?

I was trying to edge Matt towards the conclusion that "differently" has nothing to refer to in this case.
 

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