Some time ago, I read an article about Kurt Gödel and Hilbert. The article was about whether Mathematics are a formalistic Science or a platonic Science.

Formalistic means that Math is an invention of the human only present as abstract thoughts in our minds, with no relation to the nature. The consequence is that everything can be proven, it just depends on the axioms you are using.

Platonic comes from the Greek philosopher Platon. He was convinced that Mathematics are a part of the Nature and can also be found in it. As a consequence, there can be certain things that we know they are true but it is impossible to prove it.

Although Gödel seems to have proven that the formalistic point of view is actually wrong, I still would like to hear a few comments about it from others. Whast do people think in general about Mathematics? Is it a part of our nature or just a weird, complex and abstract invention only present in our minds?

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Beauty is the first test:
There is no permanent place in this world for ugly mathematics.

All of the abstract structures that make up mathematics exist independent from our reasoning on them. Our number system is arbitrary, although to most it would seem impossible to do math without it, it really has no substantial impact on any mathematical properties. A whole universe of math was made available to us when we noticed 1+1=2. Everything we have discovered (and will discover) since the axiomization of math has been the result of observable logic. Our idea of math is how we describe those axioms and the deductions that follow from them. But regardless of how we describe the axioms, that same mathematical universe will always be waiting for us to discover more of it.

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[ and creates a universe for them to describe? ]

This is more or less the same opinion as I have. But you said that the axioms are set up by us. Couldn't it be that there are axioms so fundamental that they even exist in nature and which are always true, independant of how we observe and interprete the nature around ourselves? And what could these axioms contain?
They probably cannot contain anything about the numbers themselves, since the number itself is an abstract idea, representing any given thing around us. An invention of the human, only there to help him visualize his problems and so far hardly defined at all.

But I surely they cannot contain any idea about operations ore certain structures since they depend too much on axioms only inserted later by the humans.

This looks much like René Descartes' search for his Cogito. Looking for something so fundamental and undoubtly true that it must exist. And if it exists all what follows must exist as well.

If we follow that reasoning but into the opposite direction, we can conclude, that independant on the point where we start. We will always find a rule or an axiom telling us that our starting point really exists. If we move on to that rule or axiom, we may find another axiom telling us the existence of our present place. Now following that way, are we going to come to an end? A starting point from which everything in Mathematics is coming from?

Maybe it's even as simple as the cogito. Mathematics can be seen, so there is no doubt that they really exist. And if they exist everything that seems natural and clear to us is true as well. It would suit quite good into the platonic philosophy of Mathematics because it gives us an argument against the formalism: there is at least on starting point which must be true but cannot be proven. It is true by its existence, otherwise everything would instantly vanish...

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Beauty is the first test:
There is no permanent place in this world for ugly mathematics.

Forgive me for entering a discussion that I may not be qualified to comment in however it is so interesting I can't help myself . . .

If you consider for a moment Euclid's geometry, which seems natural and clear to us and has been very useful throughout the centuries; though it is logical . . . it does not exist. Because space is not flat (einstein's theories suggest that mass bends the fabric of space) in fact none of Euclidean geometry exists in our world. Of course there is non-Euclidean geometry but the point is that for centuries people could point to geometry and say this is real . . . this exists, yet it was only an approximation to help us make sense of the physical world. Almost like religion has helped man make sense of the world though it does not explain everything and does not always work/make sense . . . so if you follow my logic, math is not real and does not exist independantly of our minds though it makes our lives meaningful and I could not imagine living without it (the same could be said for the internet ).

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Math Help

Everybody is free to comment, you are probably not more and not less qualified to enter the discussion than anybody else. Anyway we are here to exchange our opinions, aren't we?

Are you sure that the Euclidian geometry does not exist? I think it depends very much on the scale you use to contemplate an object. If we are speaking in terms of astronomy, yes you are right. Einstein has proven that space is not flat anymore and that we cannot apply Euclide's Geometry.
But what if we contemplate object at the atomic scale? Your pencil is made out of carbon whose atoms are aligned in perfect rows. The rows are aligned in one single plane and all the planes form a space. If you wanted to calculate the distance between the atoms, you will find yourself inside Euclide's geometry.

The question isn't probably whether it exists or not, but more when it exists and most of all where?

When Newton found his Gravitiy theorems he thought it was right. It took a long time before Einstein proved that it is not always right and he proposed another one. But we still use Newtons formulas for many things because their accuracy is high enough for getting useful results in a rather short period of time. Once again both are actually correct (they both can be proven with experiments) but once again it depends on the scale you want to contemplate your problem.

For your comparison of Math with Religion, I can't agree with you completely. Religion has a slightly different purpose than Math, although this again depends on the Religion you believe in. The basic idea behind Religion is to give you something that gives you a certain comfort, something which helps you to come over difficult times. Many people who have lived in a war, only survived because of their belief in Religion, which gave them the hope that everything is going to become better again. It actually gives your life a purpose, a goal to aim at. If you followed your life according to the religious rules, you will find sooner or later complete happiness. People then feel relieved because their life is not senseless.
The purpose of Mathematics is in my eyes slightly different. Although it can provide you happiness as well and give your life a certain goal, isn't its true intention not to find the absolute truth behind the things. Many Religions are based on texts which you should not question. These texts are considered to be true, even though no prove is given.
In mathematics you are welcome if you question everything. You shouldn't use things in Mathematics that aren't proven and if you still use them, you are invited to prove them. Therefor Mathematics can't give you the feeling of security that religion provides you. Everything around you could be false, even if someone brought a proof. (How many proofs did already appear for the Riemann hypothesis? They were all wrong...) One little detail which is missing and your whole universe could fall into parts.

This doesn't mean that you can find happiness throughout Mathematics. Is very something more wonderful than the feeling you have, finding that your solution to a problem is actually true?

Although this difference between Math and Religion is really small, it's still quite important.

__________________
Beauty is the first test:
There is no permanent place in this world for ugly mathematics.

Of course Euclidean Geometry exists: there's a large chunk of it in a book on my shelf at this very moment (it's Heath's excellent translation of Euclid ). The question is, does the mathematical construct of Euclidean geometry exactly model the real world around us. The quick answer is no; the next quickest is no, but it's a very good approximation. Another answer might be that the curved space which we believe we see around us can itself (under suitable conditions) be modelled within Euclidean geometry. One of the achievements of 19th century geometry was to get away from the rather sterile arguments about whether, or which, geometry was 'real' --- let's not throw that away!

Another question: What is the exact model of the real world around us? Is it real a space describe as by Euclid or is it rather more spherical? Maybe it's even total different and yet we are just not capable of seeing how it really is.

David Hume once concluded that as long as we can not sense something, we do not know that it's really there, but it doesn't mean that it doesn't exist.
Maybe the space around us is so complex around us that none of our geometries could give any adequate approximation, but just don't realize it.

We must first look for what the world around really looks like before we argue about which geometry we apply.

__________________
Beauty is the first test:
There is no permanent place in this world for ugly mathematics.

It is a good aproximation if there are no great gravitational fields in vincinity of point (near black hole for example).

My opinion is that math by it self is a great, and only objective tool that helps us observe a world, but we can't find the mathematical theory that describes exactly how world functions. That is becoase our knowledge about the world is based on experiment, and then the experiment is ilustrated by function that best fits experiment, but only in rage that experiment is conducted. Of course, you can extrapolate your function, but in extreme conditions that function fails to explain some phenomena's. For example... Gallileo and Lorentz transformations ,or Newtons and Einstein theories, or clasical and quantum mehanics. The fact is, it is imposible for us to find the exact function that will describe the world, but it is posible to get a pretty good fit in some rage in which we conduct experiments.

Sorry to go so far back in the thread on this one but I think there is an important point to make. Newton's formulas are always wrong at slow speeds they are very good approximations. Einsteins formula's are always right (until someone comes along with something better) they work at slow speeds and at high speeds. And this was exactly my point, for centuries (and still today) newton's methods were beleived to be accurate or true but in fact they were only approximations.

The real question here is does mathematics exist without us. And I beleive that it doesn't. Just as time did not exist until there was a universe. The more I think about it the more parrallels() I see between religion and mathematics. Mathematics exists in theories that are built from axioms or postulates and those postulates, though logical, do not exist without us. And as we found one of Euclid's postulates to be questionable (The one that when negated you have hyperbolic geometry) so too is the science of mathematics. Euclid (and others) did not find geometry but rather, created it. Man did not find God, but created it.

I would love to read further reasoning on how/why mathematics is a platonic science.

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Math Help

You are right when you say that the axioms and theorems that we create cannot exist without us, because we created them. But actually. What is a theorem or axiom really? It is an interpretation of an observation made somewhere in the nature. Mathematics are out there in the Nature, all we have to do is find them. We find them by observing everything around us and putting it into theorems and axioms which give us an useful method to work with them for future observations. The axiom cannot exist without the Mathematical idea present in the Nature, but the idea can exist without the axiom. There are surely theorems out there that haven't been discovered, but it doesn't mean that they do not exist yet. We all know that 1+1=2. If nobody had found out that, it would still be true that 1+1=2 even if nobody would know. Each time we calculate something, we calculate according to all the laws, even those that have not been discovered yet. When Euclid gave us his geometry he did not create it. He gave us a tool that was able to reproduce what nature had told him.

__________________
Beauty is the first test:
There is no permanent place in this world for ugly mathematics.

I disagree,

1+1=2 is entirely made up. If we did not design math in this way it would not have been self evident in nature.

When looking at the way life begins it takes 1 man and 1 woman to produce 1 child. Sometimes there are twins 1 original fetus turns into 2 babies. 1+1=?

Take a worm and cut it into two worms. 1 worm divided by two does not equal 2 halves of a worm it equals 2 whole worms. Each half becomes a whole worm. 1 / 2 = 2

I can take a twig from a tree and plant it and I get another tree. ?=?

I am sure if I spent more time on this one I could come up with some more mathematical equations that exist in nature but make no sense in the mathematics we have developed.

Further more Plato argued that we on earth have never seen two equal things. No two stones or logs are exactly the same and therefore even the idea of equality one of the basic axioms that mathematics is based upon is not evident in nature. Plato used this argument to explain how we must learn things before we are born into this world. That there is some knowledge that is impossible to extract from our surroundings.

I disagree with Plato about not being able to come up with the idea of equality from our surroundings but I agree that nature itself does not have equality evident in it.

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Math Guru

Maybe you look too strict at the equations anfd there equivalent in the nature.
You will never find something in the nature that behaves according to 1+1=2.
If you consider 1 stone and another stone you get 2 stones. Then it works 1+1=2.

1+1=2 is a human invention.We set up the decimal system and decided that 1+1=2. This isn't even true anymore when we change our number system. In binary system 1+1=10.

Numbers are an delicate question, because the problem is that they actually just represent a certain quantity, known or not, but for themselves a number is a extremely abstract thing. You can define certain sets of numbers but finding a definition for a number itself is merely impossible. So the numbers which we use in a representative way for those quantities have appeared long after the quantity itself.

Finding Mathematics in the Nature, means looking for reasonings without any nubers or equations. It is probably more a philosophical quest than a mathematical one. Although both things are sometimes difficult to distinguish.

__________________
Beauty is the first test:
There is no permanent place in this world for ugly mathematics.

Elementary particles are exactly the same.

Well I'm going a rather long step back in this discussion, but I thought that this would interest some people. It's an explanation by analogy (not mathematical, which makes it rather easy to understand) of Gödels incompleteness Theorem which actually shows that we can't prove everything.
It shows that there cannot be any ultimate and final axiom that regroups the whole of Mathematics. You will always find one or another thing that cannot be proven.

The site where I found this is: http://www.miskatonic.org/godel.html

__________________
Beauty is the first test:
There is no permanent place in this world for ugly mathematics.

Problem lies in posible solutions... the program has to make a statment that is not true. Its like asking "what's the solution for x and y in following equation:
x+y=1, x+y=2?", and you can't say that that problem does not have a solution (which is the only correct answer). Similary, with Godels example, mashine can't answer in terms of statment true/false, something like asking a man "is the sky green or purple?", and not having an option to say that the question doesn't make sence.