Mathematics- invented or discovered

take everything

Was reading a thread on another forum about the nature of maths.
This guy proposed the notion that reality is essentially mathematical. That man has merely discovered maths, not invented it.
He sounded like a crackpot but his argument was intriguing.

Dunno if this belongs in the philosophy forum but i would have thought there'd be some interest here. Is there anything in this?

This is the link (starts on the 2nd page of this thread ).

http://www.bautforum.com/showthread.php/89479-Origin-of-universe-is-humanly-incomprehensible/page2?

Yeah, there's quite a lot of it. The view that that guy takes is called "Mathematical Realism". There are lots of other views that have been written down, torn up, refined and written down again by philosophers.

There is a comprehensive Wikipedia page on all of this, and any Google search including the terms 'maths' and 'philosophy' will give you brain food to think about.

Personally, I think that mathematics as it is today can exist on its own, but that subsets of it do correspond to our universe. After all, the origins of mathematics came from counting and geometry, which were based very much on the world outside of our minds.

take everything

Deleted User wrote: »

Yeah, there's quite a lot of it. The view that that guy takes is called "Mathematical Realism". There are lots of other views that have been written down, torn up, refined and written down again by philosophers.

There is a comprehensive Wikipedia page on all of this, and any Google search including the terms 'maths' and 'philosophy' will give you brain food to think about.

Thanks.
That stuff looks like heavy going.

Eliot Rosewater

Could anyone recommend a book about this topic? I'm allergic to Wikipedia (I think it's crap) and the sources list at the bottom is too large and unpredictable to sift through (classic Wiki information-overload!). Cheers for the link nonetheless conorstuff!

FoxT

I would suggest

http://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/dp/0195130871/ref=ntt_at_ep_dpt_2

I have not read this book, but have read others by the same author, & found him readable, stimulating, enjoyable, but not heavy.

OTOH, If you have already read & understood the wikipedia article, he is probably not what you are seeking....


-FoxT

KnifeWRENCH

Funny seeing a thread about this; my final exam of the year (4th yr Maths & Physics at UCC) was History of Maths and our lecturer put down "Is mathematics discovered or invented?" as one of the questions.

My initial reaction was to say invented, but when answering other questions I mentioned <mathematician> discovering <theorem>, so would have contradicted myself if I had answered it like that.

I took the simple way out and just didn't answer that question! (The benefits of having choice of questions in an exam!) It's a fascinating question but not one I was willing to gamble marks on in an exam. :pac:

Permabear

This post has been deleted.

Fremen

This is what bothers me about philosophy - everything's so ill-defined. If you want to ask the question "does mathematics exist independently of human thought" in a meaningful way, you need to clearly define what you mean when you say "mathematics" and "exist" (and "human thought", but let's take that as given).

There are enough shades of meaning in both terms that one could probably argue either way quite convincingly. For example, the question "Does beauty exist independently of human thought" seems a lot less clear-cut, but a little reflection shows that the two questions are actually pretty similar.

I think most philosophers would define the question to their liking and then discuss it. For instance, here's one valid question I could think of, better defined than the previously mentioned one.

(Although I have no idea why anybody would bother trying to answer it)

"In mathematics, we assume axioms for arithmetic and prove results based on these. We then define a correspondence between these numbers we've defined, and the size of collections of objects in real life. Our observance that the sizes of collections of objects in real life observe the same axioms as those for arithmetic with numbers, however, is based on empirical evidence. How can we be sure that our inference based on this empirical observation will always be true?"

hivizman

Deleted User wrote: »

I think most philosophers would define the question to their liking and then discuss it. For instance, here's one valid question I could think of, better defined than the previously mentioned one.

(Although I have no idea why anybody would bother trying to answer it)

"In mathematics, we assume axioms for arithmetic and prove results based on these. We then define a correspondence between these numbers we've defined, and the size of collections of objects in real life. Our observance that the sizes of collections of objects in real life observe the same axioms as those for arithmetic with numbers, however, is based on empirical evidence. How can we be sure that our inference based on this empirical observation will always be true?"

One of the philosophers who considered the relationship between "formal mathematics" seen as a process of logical deduction from axioms, and "empirical mathematics" seen as observations about the world was Ludwig Wittgenstein (principally in his Remarks on the Foundations of Mathematics. Wittgenstein actually denied that there was a difference between these understandings of mathematics, but some empiricists would follow Hume in arguing that, even if it has been found that a particular axiomatisation of arithmetic has in the past generated theorems that correspond with observed regularities in the world, it is not possible to argue, using induction, that such correspondences will always be found.

An analogy would be with Euclidean geometry, which many mathematicians up to the nineteenth century believed was not only a complete and consistent axiomatic system but also corresponded with the physical world. However, the development of non-Euclidean geometries, and arguments that these provided closer correspondences with the world, undermined this belief.

Justin1982

You might find Max Tegmarks theory interesting:

http://space.mit.edu/home/tegmark/toe.html
http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.4024v1.pdf

Personally, after studying Theoretical Physics and dipping my toe only slightly into the Unified Theories of the Universe I was a bit unsatisifed by the undefined relationship between maths and the universe. Why does the unoverse obey all these mathematical equations and why does it act so logically?

A few years after I graduated I noticed the front page of American Acientific had an article called "What the Universe is really made of". This guy Max Tegmark had a theory that the universe is nothing but mathematics at the most fundamental level.
Personally I think he has hit the nail on the head, although I am less certain that anyone will ever be able to prove too much of it. And yes I understand that its borderline religion material (which I detest)

And maybe you might also be interested in Godels Theorems.......**** hit the mathematical proverbial fans when he came up with this stuff.

Maths is just built on basic axioms which you can set up as you like.
1 + 1 = 2 When your counting oranges
1 + 1 = 1 or 2 or 3 or 4 or 5 .....If your talking about human reproduction
The maths after that depends how you build your axioms like the above. So in that way its made up.

Enkidu

Permabear wrote: »

This post had been deleted.

Is it that simple though? For instance one could argue that no perfect circle exists and hence
there may be no naturally existing ratio which is equivalent to the mathematical notion of pi.
Also the question is simple enough when you talk about 1 + 1 = 2 and pi, but what about manifolds and groups?
Does SO(4) (the group of four-dimensional rotations) exist?

My own naive thinking is that mathematics is somewhat like chess. In chess bishops move diagonally
and knights in an L-shape on an 8x8 board, as we all know.
So, one could ask the question "Can a knight capture a bishop?", which is essentially a theorem of
chess. A proof by inspection (play a chess game) demonstrates it.

Now, is it a fact indepedent of humans that a knight can capture a bishop? Yes and no. The game of
chess is created/made up by humans by the action of listing its axioms (rules). However within those
rules it is an objective fact (independent of humans) that "knight captures bishop" can occur.

A better, but more technical example, is Zermelo-Frankel set theory and measure theory. Is there
a subset of the reals with no measure? Well if you accept the axiom of choice, then yes. However if
you accept its contradiction then no. To my mind this means a human being is free to play in two
different set theories ZFC(ZF + Choice) and ZFNC (ZF + Not Choice). The existence or not of an
unmeasurable set is an objective (discovered) fact within this two invented theories.

I guess this would make me an objective formalist.

FoxT

I think there is a mix of invention & discovery going on. The language of mathematics - the symbols etc. - were invented.

In my view, (And I know there will be many that disagree) The models that math uses (numbers,angles, vectors, for example) were also invented. However, their properties were discovered. I'd also argue that many transforms were invented.

As an analogy, I would say that Hamlet, the Mona Lisa, The Golden Gate bridge, the internal combustion engine, the bunsen burner, the electric light bulb, the Laplace Transform, the Fourier Transform, and boolean algebra - were INVENTED.

In each case the designer had an inspiration to create an object with certain properties. Just as the Mona Lisa didn't exist until it was created, neither did the Fourier Transform.

OTOH, I think that Godel DISCOVERED his incompleteness theorem. He started off with something that already existed, ( systems based on axioms) and he discovered that it had certain properties. This applies also to Fourier - who invented the transform (OK I know Fourier did not achieve this on his own & a lot of his ideas needed cleaning up afterwards by Dirichlet & others..) and others DISCOVERED that this transform had 'really useful' properties.

I don't think the question has a black & white answer. And, while it is an interesting enough question in its own way, I don't think it is that important.

I'll sign off with a quote...
"The fundamental cause of trouble in the world today is that the foolish are cocksure whilst the intelligent are full of doubt"
- Bertrand Russell



Cheers,

-FoxT

take everything

FoxT wrote: »

I'll sign off with a quote...
"The fundamental cause of trouble in the world today is that the foolish are cocksure whilst the intelligent are full of doubt"
- Bertrand Russell

Cheers,

-FoxT

Thanks for the post.
I love that quote.
How does it apply here though

Enkidu

Actually does anybody here have an opinion on the continuum hypothesis? I'm never really sure if a set theory makes more sense if it's false or true. (Of course it is neither in ZFC).

FoxT

Quote:
Originally Posted by FoxT
I'll sign off with a quote...
"The fundamental cause of trouble in the world today is that the foolish are cocksure whilst the intelligent are full of doubt"
- Bertrand Russell

Cheers,

-FoxT

Thanks for the post.
I love that quote.
How does it apply here though




Ah, looking back over it I wrote it poorly - The invented/discovered debate has been going on for a long time & there are fundamentalists on each side.

I threw in the quote because for me, the invented/discovered is not a black & white issue, and while I don't know whether Russell would have agreed with me or not, his quote seemed to fit.

Thanks!

FoxT

Enkidu wrote: »

Actually does anybody here have an opinion on the continuum hypothesis? I'm never really sure if a set theory makes more sense if it's false or true. (Of course it is neither in ZFC).

My understanding of the continuum is that

"it is not possible to construct a set whose cardinal number lies between the cardinal number of the set of Naturals & that of the set of reals"

Intuitively, it sounds reasonable to me.

If such a set were to be found, would that be an invention or a discovery?


-FoxT

Capt'n Midnight

Maths is a mixture of both. But even without examples from the physical world to try to model we could have generated maths.

In may cases what was once thought to be abstract , pure research maths has turned out to just the thing when people needed a solution to a real world problem. From modeling to encryption.

One of the reasons the Russians beat the Americans into space is that they had better mathematicians, but without computers you are reliant on fairly advanced differentiation to handle stuff like changing air resistance, changing thrust, changing weight, changing gravaitational attraction depending on height etc. But the techniques didn't need to be developed.



Some stuff like the behaviour of natural numbers is waiting to be discovered, maybe. Stuff in other planes in abstract geometiries is invented IMHO.