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Completeness Theorems of Model Theory, a branch of Mathematical Logic. Together, these two Theorems show that: under the Field Axioms (the rules of the game for scalars) existence of rationals is provable, but existence of irrationals and complex numbers is neither provable nor disprovable ,Now I want whethere exist such new theorem in philosophy which prove or disprove existence of complex quantities and irrational numbers in nature ?

  • "Existence" is a very ambiguous word, and its meaning in mathematics has little to do with "existence in nature". Existence of irrationals is provable using some very weak continuity axioms about line-circle intersections. As for complex numbers, all one needs is the ability to form pairs of reals. None of this has anything to do with nature, even positive integers are just mathematical abstractions. – Conifold May 2 at 2:41
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    The question of what "existence" means and under what conditions an entity exists is very legitimate. Somebody with time and knowledge could give a brief excursion from Plato's different realms of existence to Quine's holism where "existence" means the same under all circumstances. – Clyde Frog May 2 at 6:54
  • In what sense "existence of rational is provable" ? We can build the rationals from the natural; thus, IF natural numbers exist, then also rationals exist. This is not exactly what we usually mean with "numbers exist in nature". – Mauro ALLEGRANZA May 2 at 7:39
  • IMO, the question "Do numbers exist in nature ?" is simply wrong. Does money exist in nature ? Do states exist in nature ? Do boards of directors exist in nature ' Obviously not: they are human and social constructions. But this means that they are not "real" ? No; do you think that your bank account is not real ? – Mauro ALLEGRANZA May 3 at 9:19
  • Two questions are relevant. First one: what kind of existence do numbers have ? Are they "only" the meaning of number-words ? Or are they some sort of abstract objects ? Second one: applications. What guarantee the application of mathematical concepts and structure to reality (starting from the obvious practice of counting) ? – Mauro ALLEGRANZA May 3 at 9:26
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Irrational numbers were discovered by Classical Greeks in simple things like the ratio of sides in a 45 deg right angled triangle, which is the square root of 2. In some cultures it was declared heresy to admit such numbers were irrational and the advocate was put to death.

In an electrical resonant "Tank" circuit, the voltage and current are 90 deg out of phase. Mathematically we model this using complex numbers, with real voltage and imaginary current. I once set up such a circuit with a period of around half a second and moving-pointer meters wired in. I happily sat and watched as the needles oscillated to and fro, 90 deg out of phase with each other. The ammeter thus demonstrated that the mathematically imaginary current was in fact as physically real as anything else.

So these kinds of number clearly reflect the realities of nature every bit as much as the rationals do. Your theorem results are confined to what may be called discrete mathematics. To deal with continuities, you must add one or more "axioms of continuity" to your formulation. A similar issue occurs with projective geometry, to which axioms of continuity - among others - must be expressly added before one can apply homogeneous coordinates and hence analytical methods.

But all this is really a mathematical issue. Philosophers can do little more than ask why these various axiom sets are necessary.

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  • Complex numbers are useful in physics, but in quantum gravity there is a possibility that spacetime is discrete (suggested especially by the Bekenstein bound that is likely to be incorporated into such a theory), in which case it might indicate that the use of irrational numbers is just an idealization, similar to how classical physics may assume perfectly continuous matter distributions even though we know in reality matter is made of discrete particles. – Hypnosifl May 2 at 19:33
  • Even rationals and reals become inappropriate when you have virtual particles and mere probabilities ruling the roost; most quantum probabilities are irrational. Then again, how many rocks do you have after a couple of flakes are frozen off one in a cold winter - one or three? How big must a flake be before it becomes a rock in its own right? I'd suggest that such issues are too far off the question to be relevant. – Guy Inchbald May 3 at 9:13
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Numbers (of any kind) do not exist in nature, but in our mind. Numbers are the representation of boundaries between parts, and such boundaries are always subjective.

Written by a kid [1]:

-The dog has the following: tail, 1; paws, 4; ears, 2; eyes, 2; muzzle, 1; teeth, 32. Total: 42.

Now, think, is this true? Is that false because you are an adult and that is written by a kid? Which is the number that "exists" in nature? For what you would say, you cannot count complex numbers in the dog. But probably, a QM scientific can. Which is the truth?

[1] Jose Maria Firpo, "Qué Porqería Es El Glóbulo", Ediciones de La Flor, 2005, Argentina.

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    In what realm numbers exist and what existence means anyway has been up to debate for 3000 years. Don't think so many clever minds would have pondered upon it if numbers simply "do not exist in nature, but in our mind". – Clyde Frog May 2 at 6:49
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    Perhaps nature only exists in our mind. Plenty of folk have speculated on these lines and some insist on it. Hence many clever minds have pondered on it. – user20253 May 2 at 8:28
  • Precisely. Empiricism is the essential reference (Locke, Berkeley, Hume, and, not properly an empiricist, Kant). No mature form of philosophy isn't based on such premises. – RodolfoAP May 2 at 23:03
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I am hardly as 'deep' WRT the philosophy of mathematical logic as many here. That said ..

I am not sure what 'numbers existing in nature is', exactly. But it is interesting that the best 'representation of how things are' that we have so far is quantum mechanics. Until you get to that tricky 'observation part' it is exact (as far as we know) and I would think (but do not know how to prove) that irrational numbers would have to be part of the solution set.

My initial thought was that the use of complex numbers had to be part of that representation of 'how things are', but the Schrodinger formulation is not the only one that works.

dave

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I'll try to reconstruct your argument to show where it "goes wrong":

under the Field Axioms (the rules of the game for scalars) existence of rationals is provable, but existence of irrationals and complex numbers is neither provable nor disprovable ,Now I want whethere exist such new theorem in philosophy which prove or disprove existence of complex quantities and irrational numbers in nature ?

So, if I'm not mistaken, the reasoning behind the question is:

  1. Rational numbers can be deduced from the axioms for scalar fields, but irrational and complex numbers cannot be.
  2. If a number cannot be deduced from the axioms for scalar fields, it does not exist in nature.

C. Therefore, rational numbers exist in nature but irrational/complex ones do not.

And then you ask whether there is a philosophical theorem that can supplement the theorems used in the background of the relevant field axioms, such that this theorem would go on to allow for a relevant deduction of irrational/complex natural numbers. So there is a meta-claim that seems like:

If a theorem T satisfies the completeness theorems of model theory, then T is relevant to what exists in nature.

I suppose the intuition behind this might be that model theory, as a branch of mathematical logic, might be ontologically fundamental in some relevant way, but I feel like there's a lot missing, here, with which to make the actual case (behind the question). In other words, I don't see why only numbers deduced from the axioms for scalar fields would be the ones to exist in nature. I would think that the proper approach would be to study nature itself, and then infer what numbers "exist in it" from the results of this study. Now at least by ordinary enough intuition, we can note continuous functions in nature, wherefore the numbers of the continuum (including irrationals and complex ones) are all encoded into it. Whether there is a deduction of the continuum from some countable model of logical reasoning seems to be an unnecessary aside.

[In case it's relevant to what you're saying, I'd say you might want to look up the subject of infinitary logic, for which there is an excellent (if esoteric!) article on the Stanford Encyclopedia of Philosophy. I can't testify as to how much it involves the kind of model theory you're referring to, but I can at least testify that it addresses the question of which ordinals characterize the most useful infinitary language, namely L(ω1, ω) (where "ω1" is the first uncountable ordinal, which assuming that the Continuum Hypothesis is true is also the initial ordinal for the cardinality of the continuum); so perhaps the fact that L(ω1, ω) allows logical conjunctions and disjunctions to be < ω1 in length "shows" that the logic of a physical universe, modeled by this L, would be related to the numbers of the continuum in the way you would need to "prove that they exist in nature"; but all this is so vague and programmatic that I don't want to totally commit you to looking for an answer to your question, in this direction.]

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