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From the first day that I entered college, I was wondering about the relationship between some basic mathematical abstract concepts and nature. I'm going to explain them here and you may find them a little bit ridiculous, but please guide me in case you have good answers/examples.

Infinity: Is the infinity concept exists in reality? We all know it is an abstract concept but what I'm curious to know is that if there are any physical phenomena out there that can show, stimulate or somehow help us understand the concept of infinity in reality/nature/physical world.

Zero: What about Zeno's paradox? In nature (our physical world) there is a "smallest distance". It's about $1.6 \times 10^{-35}$ meters. Another example would be when someone says "there are three apples on the desk, if you take all of them there is $0$ apple on the desk". Obviously, it's an abstract concept, but what I would like to know is that if there is any observable physical event that can anyhow show us the concept of nothingness or absolute zero. i.e we don't have absolute zero temperature in thermodynamics, absolute zero distance between 2 points in mechanics, or absolute zero gravity in a given space etc.

Empty sets: One can imagine an empty set in nature as an absolutely empty box. But I can see it the other way around: the element "nothing" is there! So, the same way one can say the element "Nothing" is a member of a set. Obviously, it's an abstract concept and this may look like a play on words, but it is a struggle for me to find a physical counterpart for it.

$\mathbb R$ ?! (the set of real numbers): There are millions of mathematical theorems which based on $\mathbb R$! I wonder if there are any non-countable physical phenomena in the real world/nature?

Again, I know these mathematical concepts are just abstractions and help us to solve real-world problems! However, I'm more interested in the relation between these abstract concepts and our physical world/nature.

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    Michel, I am currently reading [this book](http://www.amazon.com/Everything-More-Compact-History-Infinity/dp/0393339289/ref=sr_1_1?s=books&ie=UTF8&qid=1293397472&sr=1-1) which you might find interesting, it covers a significant amount of ground with respect to infinity, and is easy to read. If there's anything specific that confuses you about our answers, feel free to leave the comments, and we will respond to them.2010-12-26
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    InterestedGuest: many thank for your link. I'm thinking about your and Yuval answer. I'll comment on them very soon.2010-12-26
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    @All: there is also a discussion thread on Meta: http://meta.math.stackexchange.com/questions/1456/is-mathematics-based-on-lies2010-12-27
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    Do I exist? Do you exist?2010-12-28
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    @Carl: What do you mean ?2010-12-28
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    A closer question is does 1 exist? Does 2 exist?2010-12-28
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    @Michel Kogan: I mean that if you want to worry about whether things "really exist", there are more pressing questions than whether 0 really exists.2010-12-28
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    @Carl Mummert, I think your question will take us too far, to the comparison of Platonic existence and "real world" existence. "Does 1 exist?" is better because it does not mix Platonic and "real world" existences.2010-12-28
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    What I should have said is that while the OP's question can be resolved to some extent by pointing at the difference between Platonic and "real" existences, Carl's approach goes into question the existence of real world objects, which is unnecessary.2010-12-28
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    If 0 doesn't exist there are NOT 0 hungry lions in your room right now.2010-12-29
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    Contrary to some of the commenters, I think this is a lousy question predicated on a thourough misunderstanding of mathematics. Objecting to $0$ or $\infty$ on the basis that they do not "exist" (and presence of scare quotes should alert one to how flimsy this line of argumentation is) is like objecting to Shakespeare's Hamlet on the grounds that ghosts do not exist. A fundamental category error is at work here.2011-05-19
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    To paraphrase a former US president: it depends on what the meaning of the word 'exist' is. I'm pretty sure you'd agree there are 0 hungry lions in your room, you may need to deposit some money to get it back to zero, but infinity is a different can of worms. General relativity has singularities, sure, but that may indicate a breakdown of the theory instead of a physical process. There's a big difference between mathematical and physical infinities!2011-05-19
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    @Michel: I'll answer with a simple question: What do you think, does a point exist?2011-07-23
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    This is not really a mathematics question. If you need an answer, here is one: if it bothers you, then, no, they don't exist. Now move on. :-)2011-07-23
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    If you don't like 0 or $\infty$, go ahead and do your mathematics without them. Euclid didn't have either, and he did great mathematics. I'm just not sure what you're going to do when you have to write a check for \$103. Will you write it in Roman numerals?2012-03-05
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    I really understand that guy, he have a great misunderstanding of symbol and meaning, $0$ isn't nothing is a symbol for nothing, if we follow your logic how could we write nothing?!2012-10-11
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    As a child in school, the teacher asked what you have if you take 2 dollars away from three. All agreed there was one dollar left. "and if you take away another dollar?" I was quick to say zero. The teacher asked if I meant zero dollars. I said no, you actually have zero of an infinite universe of things, zero cows, for example. So that explained both zero and infinity, in first grade.2013-02-12
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    I think both 0 and infinity are symbols to represent things, that doesn't make sense if they don't exist. Most people have issue with the concept of infinity, but most people don't have any questions when i tell them there are infinite amount of integers.2013-12-25
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    It should be noted that the Planck length in no way represents the "smallest possible length". This kind of reasoning would lead one to think that the Planck mass (about $10^{-5}$ grams) would be the minimal (or maximal?!) mass possible. Nonsense.2015-06-19
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    In what theory? About what theory you are asking? If you are asking about physical world, you should ask in physics.se2015-11-25

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