Does infinity Really Exist?

Of course as a human abstraction it exists (they even exist in different sizes as we have a lot of imagination), like almost everything can exist in humans minds. But what about real life infinity?

Now lets imagine that we would like to split an apple or a pear in an infinite number of pieces, could we do this? I don't think so, the following will happen:

Will not we end up with a collection of finite sets of atoms that actually compose our apple or pear?

We could accept that all these finite sets of atoms could be broken into pieces (please, do not try this at home), then we would end up with less & larger finite sets of the known elementary particles that form our fruit.

Doesn't this proof that there is no infinity between 0 and 1 IN REALITY ?

Infinity is only a human abstraction that has been proven very practical to solve lots of equations. But that is like unicorns, a product of human imagination. And as such I don't think they can be used to understand physics at all. Which may explain the difficulties encountered since more that a century to properly comprehend and model physics. For this we need and only need a set of mathematical concepts that DO exist in reality. Which is another story, that I keep for another Sunday. 😀

Have a great Sunday and please don't be stupid in the comments, this is serious!

#reality
#math
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This is the best news I have red in years, it made me really really happy this morning. Soon? Everyone will understand that the continuum hypothesis is invalid and this will in turn change the world.

#ontherightpath
#evolution
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Follow the link for beautiful maths....
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This is about the most important question of all mathematical questions!
I personally doubt that this is the answer but it is worth reading.
P ≠ NP?

+Alok Tiwari pointed out a new paper by Norbert Blum, which claims to solve a famous math problem. So if this paper is wrong, don't blame me. Blame him. ʘ‿ʘ

• Norbert Blum, A solution of the P versus NP problem, https://arxiv.org/abs/1708.03486.

Just kidding! Most papers that claim to solve hard math problems are wrong: that's why these problems are considered hard. Alok Tiwari knows this.

But these papers can still be fun to look at, at least if they're not obviously wrong. It's fun to hope that maybe today humanity has found another beautiful grain of truth.

I'm not an expert on the P = NP problem, so I have no opinion on this paper. So don't get excited: wait calmly by your radio until you hear from someone who actually works on this stuff.

I found the first paragraph interesting, though. Here it is, together with some non-expert commentary. Beware: everything I say could be wrong!

Understanding the power of negations is one of the most challenging problems in complexity theory. With respect to monotone Boolean functions, Razborov [12] was the first who could shown that the gain, if using negations, can be super-polynomial in comparision to monotone Boolean networks. Tardos [16] has improved this to exponential.

I guess a Boolean network is like a machine where you feed in a string of bits and it computes new bits using the logical operations 'and', 'or' and 'not'. If you leave out 'not' the Boolean network is monotone, since then making more inputs equal to 1, or 'true', is bound to make more of the output bits 1 as well. The author is saying that including 'not' makes some computations vastly more efficient... but that this stuff is hard to understand.

For the characteristic function of an NP-complete problem like the clique function, it is widely believed that negations cannot help enough to improve the Boolean complexity from exponential to polynomial.

A bunch of nodes in a graph are a clique if each of these nodes is connected by an edge to every other. Determining whether a graph with n vertices has a clique with more than k nodes is a famous problem: the clique decision problem.

The clique decision problem is NP-complete. This means, among other things, that if you can't solve it with any Boolean network whose complexity grows like some polynomial in n, then P ≠ NP.

(Don't ask me what the complexity of a Boolean network is.)

I guess Blum is hinting that the best monotone Boolean network for solving the clique decision problem has a complexity that's exponential in n. And then he's saying it's widely believed that not gates can't reduce the complexity to a polynomial.

Since the computation of an one-tape Turing machine can be simulated by a non-monotone Boolean network of size at most the square of the number of steps [15, Ch. 3.9], a superpolynomial lower bound for the non-monotone network complexity of such a function would imply P ≠ NP.

Now he's saying what I said earlier: if you show it's impossible to solve the clique decision problem with any Boolean network whose complexity grows like some polynomial in n, then you've shown P ≠ NP. This is how Blum intends to prove P ≠ NP.

For the monotone complexity of such a function, exponential lower bounds are known [11, 3, 1, 10, 6, 8, 4, 2, 7].

Should you trust someone who claims they've proved P ≠ NP, but can't manage to get their references listed in increasing order?

But until now, no one could prove a non-linear lower bound for the nonmonotone complexity of any Boolean function in NP.

That's a great example of how helpless we are: we've got all these problems whose complexity should grow faster than any polynomial, and we can't even prove their complexity grows faster than linear. Sad!

An obvious attempt to get a super-polynomial lower bound for the non-monotone complexity of the clique function could be the extension of the method which has led to the proof of an exponential lower bound of its monotone complexity. This is the so-called “method of approximation” developed by Razborov [11].

I don't know about this. All I know is that Razborov and Rudich proved a whole bunch of strategies for proving P ≠ NP can't possibly work. So he's a smart cookie.

Razborov [13] has shown that his approximation method cannot be used to prove better than quadratic lower bounds for the non-monotone complexity of a Boolean function.

So, this method is unable to prove a problem can't be solved in polynomial time. Bummer!

But Razborov uses a very strong distance measure in his proof for the inability of the approximation method. As elaborated in [5], one can use the approximation method with a weaker distance measure to prove a super-polynomial lower bound for the non-monotone complexity of a Boolean function.

This reference [5] is to another paper by Blum. And in the end, he claims to use similar methods to prove that the complexity of any Boolean network that solves the clique decision problem must grow faster than a polynomial.

So, if you're trying to check his proof that P ≠ NP, you should probably start by checking that other paper!

The picture below, by Behnam Esfahbod on Wikicommons, shows the two possible scenarios. The one at left is the one Norbert Blum claims to have shown.
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I like people experimenting!
Testing out some experimental coloring with the Mandelbrot...

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#Fractal #Mandelbrot #Math #Art
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Adams said. “There’s much more to do. I’ll die before I’m happy with everything.” Mathematics can indeed be quite frustrating, helas! Being human is quite frustrating...
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If you like Pascal's triangle, you gonna love this.
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An encrypted secret to be revealed by a code breaking guy or by someone able to read a very old language.
ⴰⵙⵇⵇⵉⵎ ⵏ ⵜⵎⵙⵙⴰⴳⵓⵔⵜ

ⵉⴳ ⵢⵓⵔⴰ ⵉⴷⵉⵙ ⵏ ⵛⴰ ⵏ ⵢⵉⵊⵊ ⴳ ⵡⴰⴳⵎⴰⵎⵏ ⵏ ⵓⵙⵇⵇⵉⵎ, ⵉⵥⴹⴰⵕ ⵓⵏⴱⴷⴰⴷ ⴰⴷ ⵉⵙⵜⵉ ⵉⵙⵎ ⵏ ⴽⵔⴰ ⵏ ⵢⴰⵏ ⵉⵣⵣⵔⵉ ⵜ ⵉ ⵓⴳⵍⵍⵉⴷ ⵃⵎⴰ ⴰⴷ ⵢⴰⵎⵥ ⵉⴷⵉⵙ ⴰⵏⵏ ⵢⵓⵔⴰⵏ.

ⵜⵜⵡⵓⵛⵏⵜ ⵉ ⵓⵙⵇⵇⵉⵎ ⵏ ⵜⵎⵙⵙⴰⴳⵓⵔⵜ ⵏ ⵓⵙⵉⵏⴰⴳ ⴰⴳⵍⴷⴰⵏ ⵏ ⵜⵓⵙⵙⵏⴰ ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⵉⵏⴱⴰⴹⵉⵏ ⵉⵎⴷⴰⵏ ⴱⴰⵛ ⴰⴷ ⵉⵙⵎⴷ ⵜⵉⵡⵓⵔⵉⵡⵉⵏ ⵏⵏⵙ. ⴷⴰ ⵉⵣⵣⵔⴰⵢ, ⵉⵎⵙⴰⵙⴰ ⵅⴼ ⵡⴰⵀⵉⵍ ⵏ ⵜⵡⵓⵔⵉ ⵏ ⵓⵙⵉⵏⴰⴳ ⴳ ⵢⵉⴷⵊ ⵏ ⵓⵙⴳⴳⵯⴰⵙ ⵏⵉⵖ ⴰⵟⵟⴰⵚ ⵏ ⵉⵙⴳⴳⵓⵙⴰ. ⵉⵥⵕ ⵉⵙⵇⵙⵉⵜⵏ ⵏⵏⴰ ⴰⵙ ⴷ ⵉⵣⵣⵔⵉ ⵓⴳⵍⵍⵉⴷ, ⵢⵓⵛⴰ ⴳⵉⵙⵏ ⵉⵎⵏⴰⴷⵏ ⵃⵎⴰ ⴰⴷ ⵉⴳⴳ ⵜⵉⵎⴰⵣⵣⴰⵍⵉⵏ ⵏⵏⵙ ⵉ ⵉⵙⵡⵓⵜⵜⴰ ⵓⴹⴰⵀⵉⵕ ⴰⴳⵍⴷⴰⵏ.

ⵉⵙⵔⵙ ⵓⵙⵇⵇⵉⵎ ⵏ ⵜⵎⵙⵙⴰⴳⵓⵔⵜ ⴳ ⵜⴷⵡⴰⵍⵉⵏ ⵜⵉⵎⵣⵡⵓⵔⴰ ⴰⵍⵓⴳⵏ ⴰⴳⵯⵏⵙⴰⵏ ⵏ ⵓⵙⵉⵏⴰⴳ, ⴷ ⵓⵙⵏⵜⵉ ⵏ ⵡⴰⵍⵣⴰⵣ ⵏⵏⵙ, ⴷ ⵡⴰⵍⵓⴳⵏ ⵏ ⵉⵎⵙⵡⵓⵔⴰ ⵏⵏⵙ.

ⵉⵙⵄⴷⴷⴰ ⵓⵏⴱⴷⴰⴷ ⵜⵉⵖⵜⴰⵙⵉⵏ ⵏ ⵓⵙⵉⵏⴰⴳ ⵉ ⵓⴳⵍⵍⵉⴷ ⴰⴼⴰⴷ ⴰⴷ ⵅⴰⴼⵙⵏⵜ ⵉⵙⵔⵙ ⴰⵣⵡⵍ ⵏⵏⵙ.

ⴷⴰ ⵉⵜⵜⵎⵓⵏ ⵓⵙⵇⵇⵉⵎ ⵙⵏⴰⵜ ⵏ ⵜⵡⴰⵍⴰⵜⵉⵏ ⴷⴳ ⵓⵙⴳⴳⵯⴰⵙ, ⵏⵖ ⵓⴳⴳⴰⵔ ⵙ ⵜⵓⵜⵜⵔⴰ ⵏ ⵓⵏⴱⴷⴰⴷⴰ, ⵏⵉⵖ ⵜⵉⵏ ⵓⴳⵍⵍⵉⴷ, ⵏⵉⵖ ⵙ ⵜⵓⵜⵜⵔⴰ ⵏ ⵓⴳⴰⵔ ⵏ ⵓⵣⴳⵏ ⵏ ⵡⴰⴳⵎⴰⵎⵏ ⵏ ⵓⵙⵙⵇⵉⵎ.

ⴷⴰ ⵉⵙⵔⵓⵙ ⵓⵏⴱⴷⴰⴷ ⵖⵓⵔ ⵓⴳⵍⵍⵉⴷ ⴰⵀⵉⵍ ⵏ ⵜⵡⵓⵔⵉ ⵏ ⵓⵙⵇⵇⵉⵎ.

ⵍⴰ ⵉⵜⵜⵎⵓⵏ ⵓⵙⵇⵇⵉⵎ ⵉⴳ ⵉⵍⵍⴰ ⵎⴰ ⵢⵓⴳⵔⵏ ⴰⵣⴳⵏ ⵏ ⵡⴰⴳⵎⴰⵎⵏ ⵏⵏⵙ (2/3). ⵉⵙⵙⵓⴼⵓⵖ ⵜⵉⵖⵜⴰⵙⵉⵏ ⵏⵏⵙ ⵎⴰⵍⴰ ⴷ ⴼⵍⵍⴰⵙⵏⵜ ⵉⵎⵙⴰⵙⴰ ⵓⴳⴳⴰⵔ ⵏ 2/3 ⵏ ⵡⴰⴳⵎⴰⵎⵏ ⵍⵍⵉ ⵉⵍⵍⴰⵏ ⴳ ⵓⴳⵔⴰⵡ.

ⵉⵣⵎⵎⵔ ⵓⵙⵇⵇⵉⵎ ⴰⴷ ⵢⴰⵎⵓ ⵜⵉⵔⵓⴱⴱⴰ ⵏ ⵜⵡⵓⵔⵉ ⴷ ⵜⵙⵇⵇⵉⵎⵉⵏ ⵏ ⴱⴷⴰ ⵏⵖ ⵜⵉⵣⵎⵣⴰⵏ ⴱⴰⵛ ⴰⴷ ⵉⴳ ⵜⵉⵎⴰⵣⵣⴰⵍⵉⵏ ⵏⵏⵙ. ⵉⵙⵡⵓⵜⵜⵓ ⵡⴰⵍⵓⴳⵏ ⴰⴳⵯⵏⵙⴰⵏ ⵏ ⵓⵙⵇⵇⵉⵎ. ⵎⴰⵛ ⵉⵇⵇⴰⵏ ⴷ ⴰⴷ ⵢⴰⵎⵓ ⵓⵙⵇⵇⵉⵎ ⵉⵛⵜ ⵏ ⵜⵙⵇⵇⵉⵎⵜ ⵏ ⵓⵙⵎⵍ ⴷ ⵜⵏⴼⵍⵓⵜ, ⵜⴰⵍⵍⵉ ⴰⴷ ⵉⵜⵜⵡⴰⵔⴰ ⵜⴰⵏⵏⴰⵢⵉⵏ ⵏ ⵓⵏⴱⴷⴰⴷ ⵉⵥⵍⵉⵏ ⵙ ⵓⵙⵜⴰⵢ ⵏ ⵡⴰⴳⵎⴰⵎⵏ ⵉⵎⴰⵢⵏⵓⵜⵏ ⵏⵏⵉ ⵔⴰ ⵢⵉⵍⵉ ⴳ ⵢⵉⴷⵉⵙ ⵏ ⵡⵉⵍⵍⵉ ⵎⵉ ⵜⵣⵔⵉ ⵜⵉⵣⵉ ⵏⵏⵙⵏ. ⵏⵉⵖ ⴰⵙⵜⴰⵢ ⵏ ⵡⴰⴳⵎⴰⵎⵏ ⵉ ⵔⴰⴷ ⵉⵜⵜⴰⵔⵉ ⵓⵙⵉⵏⴰⴳ ⴷⴳ ⵉⵎⵓⵙⵙⵓⵜⵏ ⵏ ⴱⵕⵕⴰ ⵏ ⵜⵎⵓⵔⵜ. ⵜⵓⵎⴰ ⵜⵙⵙⵇⵉⵎⵜ ⴰⴷ : ⵉⵏⴼⵍⴰⵙ ⵏ ⵜⵎⴰⵡⴰⵙⵉⵏ ⴷ ⵓⵎⵣⵡⴰⵔⵓ ⵏ ⵜⵙⴷⴰⵡⵉⵜ, ⴷ ⵓⵏⵎⵀⴰⵍ ⵏ ⵜⴽⴰⴷⵉⵎⵉⵢⵜ, ⴷ ⵙⴰ ⵡⴰⴳⵎⴰⵎⵏ ⴷⴰ ⵜⵏ ⵉⵙⵜⴰⵢ ⵓ
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So poetic!
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