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The Ontological Elephant in the Room
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Ontology is the controversial area of philosophy that concerns what things exist and what is the nature of existence. So I ask the question in what sense do Primegrid members think prime numbers exist? Bear in mind that according to Euclid's theorem prime numbers exist that we will never find however much technology advances.  


Kronecker said: Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. It can be taken to mean that the whole numbers (and with them the primes, I suppose) exist, while more "advanced" branches of mathematics are just human conventions (hence do not exist any more than, say, the rules of chess "exist"). /JeppeSN  


I think Kronecker might have simply meant that he liked number theory more than any other branch of mathematics.
I seriously doubt an ontological difference can be argued for between integers and other mathematical objects. For one thing number theory uses many other branches of mathematics in its proofs.  


You could also imagine, as a kind of thought experiment, that an intelligent civilization existed on some remote exoplanet. What parts of our math would they know or recognize? I think many mathematicians feel that at least arithmetic including the fundamental theorem of arithmetic should be known to such a civilization. Hence, this should be "universal" math independent of biological and cultural characteristics of the human "race". /JeppeSN  


You could also imagine, as a kind of thought experiment, that an intelligent civilization existed on some remote exoplanet. What parts of our math would they know or recognize? I think many mathematicians feel that at least arithmetic including the fundamental theorem of arithmetic should be known to such a civilization. Hence, this should be "universal" math independent of biological and cultural characteristics of the human "race". /JeppeSN
I guess you could argue that the group/ring/field axioms are somewhat arbitrary and that that is what Kronecker meant. Certainly one can try dropping or adding axioms and an alien civilization might have reason to do that. But then mathematicians of both species should be able to fruitfully collaborate.
 

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I think, therefore I prime. :) I'm only divisable by 1 and myself, or I suppose maybe by an axe murderer,... he might divide me :)
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My lucky numbers 1059094^{1048576}+1 and 224584605939537911+81292139*23#*n for n=0..26  

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The supposition that "more advanced branches of mathematics are human convention" is too myopic. It ignores that geometrical objects have selfevident properties (vertices, edges, enclosed regions) which exist independently from human convention.  

Yves GallotVolunteer developer Project scientist Send message
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Ontology is the controversial area of philosophy that concerns what things exist and what is the nature of existence. So I ask the question in what sense do Primegrid members think prime numbers exist? Bear in mind that according to Euclid's theorem prime numbers exist that we will never find however much technology advances.
I recommend the book https://press.princeton.edu/titles/5634.html (or the French edition "Matière à pensée")
Personally I fully share JeanPierre Changeux' opinion and I think that mathematics mirror our brain.
The scientist says that 2^{23329780}+1 doesn't exist, it is larger than information that can be contained within the universe. The mathematician says 2^{23329780}+1 is not prime because 193 · 2^{3329782}+1 divides it!
The real world is limited but our mind is infinite like the prime numbers...  


Ontology is the controversial area of philosophy that concerns what things exist and what is the nature of existence. So I ask the question in what sense do Primegrid members think prime numbers exist? Bear in mind that according to Euclid's theorem prime numbers exist that we will never find however much technology advances.
I recommend the book https://press.princeton.edu/titles/5634.html (or the French edition "Matière à pensée")
This book is the sort of subject matter I was raising though perhaps at a more popular level. I was very into ontology about an actual year back. I was starting to work through some academic papers.
I had got into it because I had been talking to a Christian who argued for the existence of God on the basis of the existence of abstract objects. I realised at this point I did not really know what the verb "to exist" means. (Of course I also realised my friend did not know either but was blind to it.) And at that point I had to read up on ontology.
It was frustrating because although the authors disagreed I always had the feeling that they were talking past each other. Even Plato. Did Plato really believe in a Realm of Forms or was that just some sort of elaborate metaphor, because otherwise he just did not have the words otherwise to express what he believed? In other words was Plato a Platonist?
I did find a philosophy that expressed my view: Constructivism. (I give the link to a specific wikipedia page because I think the word can be used in other ways.)
So the world really exists and is not itself a product of our brains. But when we perceive it (through physical interactions) we construct a model of it. So even people who agree that there is an objective reality (as I do) cannot agree on the nature of that reality because each is looking at their constructed model of reality rather than reality itself.
Mathematics is a special case of objective reality. Mathematical objects do not in themselves "exist" (depending on what you mean by "to exist"). Rather there are necessary constraints on the way any Reality could work and Maths expresses those constraints. It is inconvenient to use Maths without being able to say that numbers exist, so we broaden the definition of "to exist" to include numbers and other things. In doing so we are constructing models of numbers in our mind. This does not mean that maths is constructed in our minds because Reality (including the truths of mathematics which are just constraints on how Reality could function) does not care how we talk about or think about it or what models of it we construct.
Personally I fully share JeanPierre Changeux' opinion and I think that mathematics mirror our brain.
As I tried to explain above in my opinion that might be true but is really only half the story.
The real world is limited but our mind is infinite like the prime numbers...
That I really cannot agree with.  

Yves GallotVolunteer developer Project scientist Send message
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It was frustrating because although the authors disagreed
Two French people always disagree :)
My problem is not that "the world really exists and is not itself a product of our brains" (I agree) and that we have to define it with a model that we call "Mathematics".
The main problem is into mathematics. They are two mathematics: arithmetic and geometry.
Arithmetic = countable sets, Peano axioms, turing machine, ...
Geometry = uncountable sets, ZFC set theory, Hilbert space, ...
I know that many people consider that ZFC set theory is well defined and that mathematics are unified but I don't.
Let x be a real number. Then it is (with probability=1) a Chaitin omega number and then we can't compute its digits!
sqrt(2), pi, e are computable numbers which is a countable set. But the real numbers are an uncountable set then a 'random' real number can't be computed.
We can define real numbers (if we define a geometrical line then with Dedekind cuts we define real numbers), but they are some terrifying objects that our brain can't "understand".
Now what is the "real world"? A huge wave function. I think that the quantum decoherence theory is correct, there is always a superposition, the wave function collapse is an approximation.
Then the real world is a huge geometrical object that can't be reduced to a finite set of independent wave functions... no integers, no cause and effect. In this world, nothing can't be computed.
If you don't build the real line from integers and rationals, how to define the number two without a mental construction?
But we see the "real world" through our brain which is a turing machine. Through this sieve we see cause and effect, logic, sets, integers. Our brain digitalizes the universe but that doesn't mean that the universe is built with numbers.
But even if our small turing machine is not able to understand the infinite geometry of the universe, it's complex enough to create integers and here we are playing with them.  


Two French people always disagree :)
I am not sure we disgree that much but then I am not French!
Our best knowledge of the Real world is through science and the best answer that has given us is the Wave function. But that is a narrow window through which to view reality. Who really knows what lies beyond that window?
It does seem like there is little reason in quantum mechanics to view a line through space as being the Real Number line. It seems to be something else below the Planck distance.
However when I say that in some sense I believe Real numbers exist I do not necessarily mean they are naively to be found in the real world.
Take group theory or any system of axioms for example. They are not saying that those axioms are truly instantiated. Mathematics simply says that if something follows those axioms then certain things follow. But the fact that if two mathematicians agree on a set of axioms they must agree on the theorems  that is a feature of reality. In fact it is a feature of reality more fundamental than charge or mass of the electron. It would be probably still be true (if it could be meaningfully stated) even if there were no electrons.
So in this sense mathematics is more true than physics except that the only existence claims mathematics makes are those dependent on axioms.
Amongst the axioms that mathematicians might come up with some form of number must surely be very early, but that does not make them any more true than any other branch of pure mathematics.
 


I wrote in my profile: "I think prime numbers are rare gems and they could help us to understand more about universe itself."
We can use prime numbers to generate/represent all the other integer ones by product.
They exist as little (compared with the infinite) unique bricks we don't really know how to use at all.
I think math is above universe because it's abstract and lets us construct models of natural phenomena (universe) too.
I guess math already exists and our mind lets us discover mathematical things, but I obviously don't know how.
 

Yves GallotVolunteer developer Project scientist Send message
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We don't disagree but I would like to show you why the real numbers may not be used in physics.
To my point of view, the most important advance in Mathematics in the last century was the decidability theory.
At the beginning of the century everything is decidable.
In 1931 Gödel proved his famous incompleteness theorems but mathematicians considered that it was an exception and that it can be ignored.
In 1936 Turing proved that the halting problem is undecidable. If you are a programmer you know that the halting problem is everywhere and that it's not an exception but mathematicians considered that programming language and mathematics are different and that we can ignore it.
In the 1980s the beautiful work of Gregory Chaitin linked the axioms of mathematicians and the Turing machine. First he built a map between the Diophantine equations and the programs of a Turing machine. Then the Diophantine analysis problem "Are there finitely or infinitely many solutions?" and the halting problem are identical. He extended his work and built the famous Chaitin omega number. It's a real number because its sum converges between 0 and 1 but it's not a computable number (there is no computable function that enumerates its binary expansion). Again mathematicians ignored this weird object.
But at the end of the century, Calude, Hertling, Khoussainov, and Wang https://www.cs.auckland.ac.nz/~cristian/approxcompactIJBC.pdf proved that a recursively enumerable real number is an algorithmically random sequence if and only if it is a Chaitin's omega number. And since Kolmogorov, we know that almost all real numbers are algorithmically random sequence.
Then undecidability is everywhere in real numbers. We have to split the real numbers into two sets: the set of computable numbers which is countable and the set of Chaitin's omega numbers which is uncountable.
If the set theory (ZFC) is free of contradiction, we can use its axioms for proofs. But can we use its "objects/definitions" in physics? Nobody uses transfinite numbers in physics but we can define them with ZFC then what about real numbers? Is it reasonable to use some numbers that we can't compute?
Some mathematicians and physicists think that we can't use real numbers and they created the digital philosophy (or digital ontology).
If this philosophy is followed then our world is built on integers and Luigi R. is right: "prime numbers are rare gems and they could help us to understand more about universe itself".
I like JeanPierre Changeux' idea that the computable numbers (or the Turing machine) is our brain. The Real world is something incredibly more complex, maybe with ZFC, real numbers, etc. But through the sieve of our mind we can just see the result of some computations: some integers. The philosophy is different because in that case the standard model is more the structure of our mind than the structure of the universe. But we still have to study integers to understand our brain and what Real world might be. I would say that "prime numbers are rare gems and they could help us to understand more about our mind".  

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Yves Gallot wrote: The philosophy is different because in that case the standard model is more the structure of our mind than the structure of the universe.
Indeed! The odd thing about quantum mechanics is the role observers have in collapsing wave functions that describe physical events.  


We don't disagree but I would like to show you why the real numbers may not be used in physics.
To my point of view, the most important advance in Mathematics in the last century was the decidability theory.
At the beginning of the century everything is decidable.
In 1931 Gödel proved his famous incompleteness theorems but mathematicians considered that it was an exception and that it can be ignored.
In 1936 Turing proved that the halting problem is undecidable. If you are a programmer you know that the halting problem is everywhere and that it's not an exception but mathematicians considered that programming language and mathematics are different and that we can ignore it.
When I was a postgraduate mathematician this is the roughly area I was working in. I also remember reading papers on mathematically interesting statements that were undecided by Peano's axioms.
In the 1980s the beautiful work of Gregory Chaitin linked the axioms of mathematicians and the Turing machine. First he built a map between the Diophantine equations and the programs of a Turing machine. Then the Diophantine analysis problem "Are there finitely or infinitely many solutions?" and the halting problem are identical. He extended his work and built the famous Chaitin omega number. It's a real number because its sum converges between 0 and 1 but it's not a computable number (there is no computable function that enumerates its binary expansion). Again mathematicians ignored this weird object.
But at the end of the century, Calude, Hertling, Khoussainov, and Wang https://www.cs.auckland.ac.nz/~cristian/approxcompactIJBC.pdf proved that a recursively enumerable real number is an algorithmically random sequence if and only if it is a Chaitin's omega number. And since Kolmogorov, we know that almost all real numbers are algorithmically random sequence.
Then undecidability is everywhere in real numbers. We have to split the real numbers into two sets: the set of computable numbers which is countable and the set of Chaitin's omega numbers which is uncountable.
I am unfamiliar with this but it sounds like a beautiful bow tied on what I did know.
If the set theory (ZFC) is free of contradiction, we can use its axioms for proofs. But can we use its "objects/definitions" in physics? Nobody uses transfinite numbers in physics but we can define them with ZFC then what about real numbers? Is it reasonable to use some numbers that we can't compute?
Yeah I don't see any relevance of ZFC to physics. Complex numbers, Real numbers, groups and so on might be relevant but the claim of relevance is only that they are models which help the calculation of predictions.
When I claim that Real numbers are in some sense in Reality that is not a claim about the physical world. It is a claim about what Realities are possible and so is a more fundamental force than physics.
Some mathematicians and physicists think that we can't use real numbers and they created the digital philosophy (or digital ontology).
If this philosophy is followed then our world is built on integers and Luigi R. is right: "prime numbers are rare gems and they could help us to understand more about universe itself".
In as much as we can't truly know what lies beyond the narrow window of our senses, this seems as plausible as anything.
I like JeanPierre Changeux' idea that the computable numbers (or the Turing machine) is our brain. The Real world is something incredibly more complex, maybe with ZFC, real numbers, etc. But through the sieve of our mind we can just see the result of some computations: some integers. The philosophy is different because in that case the standard model is more the structure of our mind than the structure of the universe. But we still have to study integers to understand our brain and what Real world might be. I would say that "prime numbers are rare gems and they could help us to understand more about our mind".
Aren't our brains too messy too organic to be the Turings machine or the actual repository for mathematical objects? Either I misunderstand this part of your ideas or I disagree with them.  


Yves Gallot wrote: The philosophy is different because in that case the standard model is more the structure of our mind than the structure of the universe.
Indeed! The odd thing about quantum mechanics is the role observers have in collapsing wave functions that describe physical events.
From what I understand this role is usually overstated. It seems that what physicists mean by an "observation" is some sort of physical interaction with the system.  

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Indeed! The odd thing about quantum mechanics is the role observers have in collapsing wave functions that describe physical events.
From what I understand this role is usually overstated. It seems that what physicists mean by an "observation" is some sort of physical interaction with the system.
Yes, the "observer" was a sort of oracle in the Copenhagen interpretation of quantum mechanics. But today the wavefunction collapse is no longer a postulate.
The "observer" are the 10^{20} particles of the sensor and quantum decoherence explains the decay of quantum information. In the air the decoherence time is about 10^{30} s, in laboratory vacuum about 10^{20} s but in intergalactic space it may take several years if the observer is a single molecule.
There is no discontinuous collapse but a "fast" exponential decay.
Theoretically the universe is a single quantum entanglement. But because of the exponential decay of some components, in practice we consider that they are null. Then we have some independent quantum states, some "particles".  

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Aren't our brains too messy too organic to be the Turings machine or the actual repository for mathematical objects? Either I misunderstand this part of your ideas or I disagree with them.
The question is "what is the best model for our brain?"
A huge finitestate machine linked to a big input device and a small output device.
The memory of the finitestate machine is so huge that the Turing machine is a model.
Roger Penrose proposed that quantum effects feature in human cognition (The Emperor's New Mind). I don't think that it's necessary. A simple cellular automaton can generate chaos (Rule 30).
A tree is extraordinarily complex but according to the physicists it's a set of protons, neutrons and electrons. If the brain is not an extraordinarily automaton, what is it?  


Aren't our brains too messy too organic to be the Turings machine or the actual repository for mathematical objects? Either I misunderstand this part of your ideas or I disagree with them.
The question is "what is the best model for our brain?"
A huge finitestate machine linked to a big input device and a small output device.
The memory of the finitestate machine is so huge that the Turing machine is a model.
Roger Penrose proposed that quantum effects feature in human cognition (The Emperor's New Mind). I don't think that it's necessary. A simple cellular automaton can generate chaos (Rule 30).
A tree is extraordinarily complex but according to the physicists it's a set of protons, neutrons and electrons. If the brain is not an extraordinarily automaton, what is it?
I agree that our brains must be Turing machines. That is the materialist position to which I hold. I thought you were proposing something quite different. (Of course there are probably better, more insightful and more efficient models of our brains. However they must be translatable back to Turing machines however inefficiently.)
Rereading this post I think the small disagreement/misunderstaning I had with you is vanishing.
You have given me some stuff to think about so I will stop here.  

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Aren't our brains too messy too organic to be the Turings machine or the actual repository for mathematical objects?
That's the beautiful part, you don't need to understand emergence for it to work. "Messy" is a human concept, describing whether we understand or can figure out how it works or we think it's "just chaos". But it does work on itself without our knowledge, emerging just from a complex system. A bunch of complex chemicals emerge into a living cell! How?! We know a lot about it, but nobody would be able to explain it in full knittygritty details. A bunch of individual cells emerge into organs and whole "living" creatures! OMG! It doesn't even stop there. A colony of ants emerges as its own entity with properties and actions that it takes, even though it's supposed to be just a "mess" of individual ants. Humans form societies and nations. All the while the "building blocks" of everything stayed the same.
Back to the brain, in latest years, neural networks with deep learning and stuff had a big boost of study and applications. It's commonplace to hear about somebody "training a network" for some task. And it works. Out of thin air, basically just a pure linear algebra is successful in some sort of pattern recognition. This is a great series on the subject, easy to digest. A model is constructed, a bunch of numbers are multiplied and added, you chose some threshold function for neuron firing, "training" is just a calculation of gradients and divergents and stuff with these numbers against known examples (i.e. other numbers). And in the end, after the calculations, this matrix of numbers, nothing but math, is able to tell written digits apart in images.
Can people, who work with this, explain, why and how it works? Not really. At all. Grant talks in the video about ideas, like, maybe neurons of that layer try to recognize certain abstract structures and then the next layer combines those structures in required combinations...
...But then you look at the visual representation of those numbers and they look just like a completely random "mess" of pixels. Without anything like those abstract structures. The researchers can tell what threshold functions work better in some way or another to train the network. But not what's going on in any individual neuron. And I don't think that'll change much in the future. It's the complexity of thinking in 10^5 dimensions. We cannot even really handle four!
This is, obviously, not The Model of the biological human brain in a reasonable proximity, but it shows on principle of what kind of turing machine or automaton, or anything it is. Or rather, how we are (or rather are not) capable of approaching it.
Those numbers were just finite precision floating point numbers. We can think of them as rational. Basically, I might be wrong, but I imagine physics doesn't care about true real numbers either. Just a subset of rational numbers that are approximating the observed values with a high enough precision "is fine". No care about computability questions, but measurement instead.
 

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Those numbers were just finite precision floating point numbers. We can think of them as rational. Basically, I might be wrong, but I imagine physics doesn't care about true real numbers either. Just a subset of rational numbers that are approximating the observed values with a high enough precision "is fine". No care about computability questions, but measurement instead.
The problem is the differentiation (Achilles and the tortoise paradoxe). Real numbers were built for derivatives.
 

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