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Part and parcel of Plato's Platonic Realism is his theory of Forms or Ideas, which refer to his belief that the material world as it seems to us is not the real world, but only a shadow or a poor copy of the real world.

Theory of Forms:

The Theory of Forms (Greek: ἰδέαι) typically refers to the belief expressed by Socrates in some of Plato's dialogues, that the material world as it seems to us is not the real world, but only an image or copy of the real world. Socrates spoke of forms in formulating a solution to the problem of universals. The forms, according to Socrates, are roughly speaking archetypes or abstract representations of the many types of things, and properties we feel and see around us, that can only be perceived by reason ( Greek: λογική); (that is, they are universals). In other words, Socrates sometimes seems to recognise two worlds: the apparent world, which constantly changes, and an unchanging and unseen world of forms, which may be a cause of what is apparent.

The material world is limited to the quantum of energy $E=nh\nu$

  1. what is the mathematical world limited to?

  2. Why are mathematical principles sometimes able to interpret the material world?

  3. What is it that is behind the material world and is sometimes connected to the mathematical principles?

  4. Why does our physical world have a mathematical structure (just like the skeleton of a building)?

  5. Did the material world create mathematics, or did mathematics create the physical world?

at the end:

Is it possible to limit mathematics to the material world (in a kind of unification)?

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I'm sorry, but this question is off topic. –  Peter Tamaroff Jan 10 '13 at 13:15
    
@Peter Tamar off i'm here to turn it on. –  Neo Jan 10 '13 at 13:19
    
Mathematically, the "Real World" is an unproven hypothesis. –  RBarryYoung Jan 31 '13 at 14:54
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4 Answers 4

The first response to a question of the sort

what is the mathematical world limited to?

is: what do you mean by "the mathematical world"? If you wish to interpret mathematics not just platonically in a colloquial sense — i.e. that there exists an absolute mathematical truth because of [insert reasons] — but take an actually Platonic position that there exists an actual world of forms, which you might call the mathematical world or of which the mathematical world is a subset (together with chairs and horses) of which our world is but a poor shadow, then it cannot be limited by our world unless you make various assumptions.

If you will forgive my injecting some mathematical terminology, you might suppose that the projection from the Platonic realm to ours is a total function: that is, for all objects in the Platonic realm, there exists an object in ours which is its shadow. This would mean that the only things which exist in the Platonic realm are those which can be identified as an abstraction of our own world. (If you take this position, you should wonder how seriously you should take the idea that the Platonic realm isn't just an invention of your imagination to categorize the structures that you already see around you.) This assumption would then allow you to try and discern some limitations of the Platonic realm imposed by representation in the domain of our experiences. But then you are stuck with problems such as: when can you identify what Platonic object is being represented by a real object? Does it matter how crude a representation the real object is, of a Platonic object? If not, then does even assumption even impose any restriction on what sorts of objects might exist in the Platonic realm?

I find that your questions 2–4 are really different phrasings of the same question, and amount to commenting on the Unreasonable Effectiveness of Mathematics in the Natural Sciences. The short answer is that there's no real way to tell, except perhaps the reasons touched on by Bruce's earlier answer: that there exists some structure in the world, certainly enough to allow self-replicating agents to arise and to form complicated social animals; that mathematics is nothing more or less than the collection of formal tools which contain — as a subset — those patterns which we see arising around us.

Basically, the question amounts to what it is that you think mathematics consists of, and where you think it came from. Not where mathematical structures in reality came from, but mathematics itself. If you think that it sort of appeared mysteriously in human culture for no reason, like some sort of divine revelation, it is not really surprising that the relationship of mathematics to the real world would be mysterious. But if mathematics — and generalizing to all subject matter of the Platonic realm, all language and logic — arose as tools made by humans to try to encapsulate not only the patterns they see in the world, but all patterns that they could imagine, then perhaps the apparent presence of mathematical structure in the world is not mysterious; it's because mathematics consists of variations on some basic themes, which we once deliberately abstracted from the physical world.

What we observe is that the physical world is not only structured, but that the complication of that structure does not vary too wildly with scale — at least, not until we consider speeds close to that of light, masses greater than that of the Earth, or spatial scales close to those of atoms. Then we find that the structure of the world is still describable by math, but — quite significantly — that it seems badly counterintuitive; and in the case both of general relativity and quantum mechanics, unweildly and difficult to compute. Is mathematics truly effective there, if we can describe the apparent pattern of things but are unable to very efficiently verify that these patterns hold true by computing their predictions? Does this represent a qualitatively robust understanding, which is different in character than an understanding of Jupiter's cloud layers by a thorough examination of the colours of its atmospheric bands from beyond its orbit?

Mathematics is useful, and is perhaps one of the crowning achievements of humanity, but it is important to keep things in perspective. Developing a psychological theory of mathematics, as with language, is important to be able to arrive at an accurate assessment of why it is as it is, why it is useful, and importantly to be reminded of its limitations, which (in some subcultures of modern society) we are often keen to forget.

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This is so good... You didn't just give a very clear answer, you even made sense of the question (that I didn't understand before). –  iphigenie Jan 11 '13 at 17:02
    
@Niel Mathematics paints a picture of the world –  Neo Jan 11 '13 at 21:32
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@Neo: poetry paints a picture of the world as well. Does that mean that the world has an inherently poetic structure? Or does it mean that we see structures in the world which appeal to us, by reason of our evolutionary development, and we celebrate those structures by depicting them or making use of similar features in poetry? –  Niel de Beaudrap Jan 11 '13 at 21:33
    
@Beaudrap:Nice answer. –  Mozibur Ullah Jan 19 '13 at 20:39
    
@Neo:The only thing I can add, and its something I was confused about, and probably still am, is Platos theory of forms essentially about mathematics? It seems to me only a subdomain of his forms, and not the whole. –  Mozibur Ullah Jan 19 '13 at 20:40
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"Material world is limited to the quantum of energy $E=nh\nu$" what Mathematical world is limited to?"

  • The meta world of human mental space

"Why sometimes mathematical principles are able to interpret the material world?"

  • Purely by a happy coincidence. Check Einstein for a discussion of this.

"what is behind material world that Sometimes is connected to the mathematical principles?"

  • The central limit theorem and the fact that what we call the real world is an interpretation by the human mind - the creator of math

"Why our physical world have a mathematical structure?(Just like the skeleton of a building)"

  • How do you know the physical world has a structure. The interpretation of the physical world by the human mind is all you/we know. Again the fact that this interpretation can be mapped to mathematical models has more to do with how the human mind works than anything to do with the physical world.

"the material world created mathematics, or mathematics created the physical world!?"

  • Humans were created by the physical world. Humans created math. Therefore, math was created by the physical world.

"at the end of both fields: is it possible to limited mathematics to the material world!? (in a kind of unification)"

  • How do you know math isn't limited to the physical world? Do you know enough about the physical world to prove that any outcomes predicated by math are not possible?

untagged

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Do you know enough about the physical world to prove that any outcomes predicated by math are not possible? no, i dont know! –  Neo Jan 10 '13 at 17:30
    
Which is a fine answer. –  Bruce Zenone Jan 10 '13 at 19:04
    
How do you know the physical world has a structure? The interpretation of the physical world by the mind is all we know.The fact that this interpretation can be mapped to mathematical models is how the mind works.Very good."The brain in the Vat" skepticism and the solipsism are interpretations or models.Any language which provide a practical way of thinking and make sense about natural laws, a common mind-independent world, must provide a way of expressing common truths about this world. What varies are the models and interpretations, but if they are to be true, what they predict does not vary –  Ricardo Jan 11 '13 at 11:35
    
BTW-I am no philosopher, though I am interested in philosophy. My comments are based on practical considerations and personal experience. Given that, they aren't worth much, but I haven't seen any of the philosophers attempt an answer. Great discussion though. –  Bruce Zenone Jan 11 '13 at 13:38
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This answer would be much improved by a development of the content beyond a ping-pong response to the idea by one-liners, with references explicitly included where possible. The fact that the responses are presented as quotations is also confusing and not really appropriate. –  Niel de Beaudrap Jan 11 '13 at 13:54
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I think mathematics has noting to do with the real world. The reason is in mathematics we start with a set of axioms and then try to explore all logical consequences of these axioms. Therefore in any possible world in which these axioms are valid and our logical thinking is accepted we will get the same mathematics.\

However physics and other basic sciences use mathematical tools (methods and theories) to develop models and theories about our real world.

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I can see why you might say that, but isn't it historically true that we derived practical mathematics from the world, and then later noticed that we could could abstract axiomatic systems from them, and then much later noticed that they could be arbitrary, but in fact generally they're not as organic growth is just as important in mathematics as it is elsewhere? Someone pointed out that Paninis grammar of Sanskrit was an axiomatic system from language. –  Mozibur Ullah Jan 19 '13 at 20:33
    
Of course, many mathematics developments are inspired by physics or other sciences. What I am saying is that the theorems and generally mathematical statements do not depend on the real world, in other words, they are certain implications that are valid even in a fantasy world (provided that we choose the same axioms and apply the same logic). So basically they do not say anything about the real world directly. But they can be used by other disciplines to create a model for the real world. –  Vahid Shirbisheh Jan 19 '13 at 20:55
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Mathematics is the study of groups of axioms, theorems deducted from such groups of axioms and models that obey those axioms.

The limits so far as we know are rather deep pertains to the distinction between syntax and semantcs, the notion of a set of all subsets and proof of own inability to contradict oneself.

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