Our answer is NO. Think of it: Pythagorean Triples would NOT exist if numbers are arbitrary as values. Given a and b, c is fixed or it doesn't exist.
Given 2 and 3, what is c?
Prime numbers seem not arbitrary either. Some people consider prime numbers as just some feature of Z, which does not exist for composite numbers. And, they think, there are no primes or coprimes in Z_p, p-adic numbers, except for some numbers, which end by 0; there are no negative and positive numbers; there are not even or odd numbers (I e., they may point to the number 19 underscore 31, is it even or odd?).
Instead, let's be humble and observe nature. A prime number in any place of the universe must be a prime number. Here on Earth and in the star Betelgeuse. It is not a feature defined by a human.
Dedekind (1888) was incorrect, and mathematical real-numbers an illusion, that cannot be calculated (Gisin, Gerck).
That is why a number is a semiotic quantity. Numbers can be thought of as a 1:1 mapping between a symbol and a value. Digits become a “name”, a reference, and it is clear that one can use different “names” for the same number as a value.
So, the number 1 can have a name as "1", "2/2", "3/3" and infinitely many more, but is always 1 in value.
Equality of rational numbers does not have to have the same name for each other, as "2/3=2/3".
They can also obey the rule that their cross product is equal in value, so that "2/3=4/6".
That way, equivalence extends equality in a consistent way, even though the numbers are neither equal nor divisible. This is possible because numbers are semiotic quantities, and is essential to understand quantum computing.
Numbers are not arbitrary as values, which can allow us to calculate prime numbers using periodicity.
What is your qualified opinion?
Basic semiotics: the last star to disappear is called the Morning Star; the first star to appear is called the Evening Star; both have different names -- but are the same planet Venus.
Numbers are a semiotic quantity. They can have any name, subjective-- but one value, which is objective.
The question about "2 and 3" what is "c" is not very well -defined.
Also, what does it mean to say "Z does not exist for composite numbers?" Does not Z consist of 0 and all the negative and positive integers? Why can't the prime factors be with respect to the absolute values? If you mean the prime factors are not unique with respect to sign, that depends on what the agreed setup is, and the absolute values of the prime factors are unique.
Please say more.
The problem with i is shown in multiplying (or dividing) as
(x+iy)*(a+ib).
These operations "contaminate" the real and imaginary parts with the other.
The equation
x^3 - 6x - 4 = 0, has its 3 real solutions in the set Q given by Eq.(1) below:
cubic-root(2 -
square-root(-4)) +
cubic-root(2 +
square-root(-4)) (1)
where the square-root is of -4, which SEEM not in the set Q, but "infest" the three results that should be in the real number line.
DH: Thank you. I will try to explain better.
1. The question "Given 2 and 3, what is c?" uses Pythagorean Triples, which say that c is not arbitrary.
2. Some people consider prime numbers as just some mysterious feature of the set Z, which does not somehow exist for composite numbers. What is the origin of the all-important prime numbers?
3. No to 2. The prime numbers are caused by periodicity, an objective property. Each digit in a prime number is no longer a digit -- together all the digits in a prime number form an indivisible "lump".
Take 311 for example. No one can take out any one digit.
So, is the familiar 3, 4, 5 a Pythagorean triple having the property that 3*3 + 4*4 = 5:5 ?
The naming of numbers is generally not arbitrary if one expects the number system chosen to be useful for counting and successful arithmetic. If the system has N as a model, or is an interpretation of N, we regard them to be equivalent schemes in that respect.
In the mid-60s a demonstration of functional programmng by Peter Landin was the function Verbalize(i) for which
Verbalize(10273) = "ten thousand two hundred and seventy-three".
I suspect it is also expected of a number system that there be canonical forms which can be enumerated and the variety of alternative forms are simply equivlent.
I agree that it is difficult to avoid confusion of the decimal numeral 10 with the mathematical 10th successor of 0 because our tacit understanding of decimal numerals leaps in. If we choose to use a different number system, it would be easier to distinguish the form of numerals from the corresponding number, except we'd probably be corresponding it via elements and arithmetic of our familiar decimal number system anyhow.
These are not arbitrary, it seems to me. Familiar or not, the key requirement being that arithmetic in such a system is satisfactory and hence corresponding with something like Peano Arithmetic.
Many researchers embrace random numbers, which deny causality, the basis of any science. Random numbers do not exist.
Many researchers affirm that numbers are whatever they want, which denies quantum computing, and factoring. Yet, it works-- creating a "wormhole" between Number Theory in pristinelly "pure" mathematics to quantum computing.
People are not being being contrarians. They are just lost, and do not see an order in nature, that is (however) always evolving ... by itself.
Nature is alive, guided by God. Trust your destiny!
DH: The form of numerals cannot correspond to their values, by semiotic reasons.
In the same way that there is no intrinsic theory/meaning of names, there is no intrinsic value for names or referents.
Names and sense are independent quantities, as well as referents. If there is smoke, there may not be fire or warmth at all.
People try to deny or resent any authority-- so much has it been used and abused.
But computers hold authority over results, and are the result of billions of people and billions of machines. Calculate the probability of one error -- it is very, very small.
Mathematics, a rudimentary creation of humans, is being rewritten by computers. Discontinuous functions can now be differentiated and integrated-- openning quantum computing.
Function values are now digital, discontinuous -- and that is more exact than a pretend continuity.
Everything that is measured, produced, or constructed is necessarily a rational number.
This is also physical, because atoms exist. We live in a quantum world, the real world.
Continuity may exist in a spiritual world, without matter.
But, if there is matter-- it is quantum, ontically.
We do not have the right intuition for a world that is purely spiritual. If it is continuous, time will tell. It might be difficult to relay back, or useful. We probably do not even have the proper words. Too abstract.
Evolution does not only affect biology, or began yesterday.
A number is not arbitrary in value -- see the prime numbers. There is nothing arbitrary about nature. Prime numbers do not depend on what humans think.
Anyone should understand that a prime number on Earth and in the star Betelgeuse are the same!
A number is a semiotic concept, and becomes an objective concept, valid for friend or foe, whatever one may want to call it.
That's why a prime number is important for physics, cybersecurity, communication, number theory in "pure" mathematics, and other areas including biology, creating multiple "wormholes " to explore it under different angles.
Mathematical real-numbers, mathematical complex numbers, irrational numbers, Lie numbers, and other man-made concepts must now be rejected, and deprecated after 500 years of false use.
The numbers to use in all sciences must be based on Boolean numbers (set B), natural numbers (set N), integer numbers (set Z), and rational numbers (set Q).
Other sets can be added, for human benefit, such as set Q*, but are not needed by computers -- that use only set B.
By the Curry-Howard relationship, this deprecates Gödel's uncertainties.
So must be finally accepted, under experiments -- not theory or opinions.
No longer individually distinguishable, the digits in each prime number is a "lump" and belong together, in a collective effect beyond digits or names. Peter Shor said this first, in 1994. This is important for quantum computing
Of course, it is agreements that have there be numbers in the arithmetic/algebraic/mathematical sense, and there are shared agreements among the practical employment of numbers as there is significant tacit understanding. That does not make them arbitrary, because there is purposeful usage and associations in a practical system. The correspondence is among seeming equivalents and agreements that have great dependability. It is not about intrinsics, it is about agreement, just as in our having a mostly-shared language, however tentative and provisional it all is on each of our parts.
We are not in 1888 any more (Dedekind). We are in the XXI century, and almost 30 years after Peter Shor.
Numbers are not arbitrary, and so help God open our understanding to outside laws. Kronecker summarized well; we can see God in numbers (cf. Ramanujan).
Quantum computing will show the rest. Mathematics is now an experimental science. Progress continues.
How about a prime number? Use 17 -- and that is another argument not in favor of imaginary numbers. We just found two different imaginary numbers that factor a prime number. We created a contradiction, as no number should factor 17.
Instead, we suggest that one should not consider the square-root of any negative number.
The preprint on the new set Q* has been updated.
In summary, a number is a semiotic concept, and becomes an objective concept, valid for friend or foe, whatever one may want to call it.
I finished my exposition for now. Yes, given 2 and 3, c is 5 -- forming a well-defined Pythagorean triple. The number c is not arbitary.
Primes, prime divisors, and the prime number theorem all apply to ℕ, not ℂ.
This misdirection is not helpful. So x^2 + 1 = 0 has solutions in ℂ, but not in ℝ.
If that's not helpful, don't use it. Whether it is valuable to others, please get over it.
DH: Others may mislead themselves and others. It has been 500 years already, of illusions and mirages, in the name of ... complicated.
Quantum mechanics, according to Schrödinger, should not use imaginary numbers. Computers do not use imaginary numbers.
I am not deciding nature. Just pointing out the inconsistencies.
There are more, to be revealed. The inconsistencies seem to invalidate Galois theory. Maybe radicals can solve any quintic polynomial in rational numbers. Do you see why not?
DH and SPK: The set ℂ (mathematical complex numbers) is condemned because it uses the set ℝ (mathematical real-numbers), which is condemned (see Gerck 2022, Gisin 2022).
One could think to still use the set G (with rational imaginary numbers), but not only the other inconsistencies apply, but every path must begin and end in a rational number, with no room for imaginary numbers.
In summary, it is time to give up any investigation. It is now an illusion.
But, I don't define nature nor evolution. I am just pointing out "not this way!".
Responding to a private message, logical constructions are similar to a popular "house of cards".
One does not have to topple it "top down", but one can go "bottom up" with more expediency.
By affirming nature as measured and constructed by rational numbers, only, we affirm quantum computing, and DENY, in sequence:
1. the mathematical Gauss set of imaginary numbers;
2. the mathematical real-numbers;
3. mathematical complex numbers;
4. random numbers;
5. infinitesimals;
6. the Cauchy epsilon-deltas method; and
7 . Gödel's uncertainties.
In addition, this logically threatens to break the support for Galois results. Other "surprises" may be in store.
Mathematics gains, as it is simplified and more effective. Humans learn to be humble, observe, and be grateful. Everyone wins.
We just need to be patient, and know that evolution will reach all. Peace and goodwill to all.
DH and SPK: Me thinks that you guys and group are just mixing contexts.
You are right when using p-adic numbers, that numbers are arbitrary. There is not even the notion of odd or even numbers. But, that is not true using B, N, Z, or Q.
Thus, whether numbers are arbitrary or not depends on the set that they are defined in, their respective context. For indigenous tribes, some do not know numbers higher than 5. The Mayans used base 20.
In the B set, numbers are either 0 or 1. And 0^n=0, while 1^n=1, so arithmetic is fast and easy. Digital computers only use the B set, and yet can calculate everything.
We, humans, can use the Q* set for fast and easy mental calculations. A negative times a negative is a positive.
Quantum computing uses the set Q, to allow calculus with discontinuous functions--as functions must be in the digital world. We see that world in the XXI century.
Your view might or not be useful with p-adic numbers, but it is certainly not universal. So, consider numbers arbitrary in that context.
My preference is what makes physical sense. In my view, C, R, and G, do not. P-adic nunbers; in my view, need to change metric in order to be trustworthy in a physical sense.
All is well in a semiotic sense.
It is odd to be seeing the same posts on two distinct topics. Here's a way I have of looking at some things concerning mathematical entities and also around computation. I do not claim that it is *the* way. I offer it as an useful way to approach some topics that I am interested in around models of computation and the utility of computers. https://orcmid.blogspot.com/2019/02/miser-project-interpretation.html
DH: With due humility, there is "the way" to calculate. Numerically, use the set B={0,1} in a computer. Correctly designed, it can mathematically reach the set N, arbitrary-length integers (set Z), and the set Q.
B → B^n → Z → N → Q. All → are due to an isomorphism, absolutely exact. Rigorous.
It also wins algebraically and logically (Curry-Howard). Mathematics becomes an experimental science.
As Voevodsky envisioned, computers can flawlessly provide numerical, algebraic, and logical proofs. Mathematical proofs finally without errors.
The way is, thus, the computer.
Computers make errors when forced to use the sets R or C, because the sets R and C have mathematical impossibilities, not because of computers. See [Gerck 2022, Gisin 2022]. That has given computers a "bad" reputation -- but is undeserved.
Calculus becomes exact when following computer mathematics, using the set Q with no epsilon-deltas, or infinitesimals, and reaching the same formulas faster. See [Gerck 2022].
The FT (Fourier Transform) becomes equal to the FFT, and much faster. See [Gerck 2022]. No trigonometric functions need to be calculated in the FT, or irrational numbers.
P-adic numbers, irrational numbers, imaginary numbers, real-numbers, and others, are to be abandoned. Keep B → B^n → Z → N → Q.
Mathematics will need to change, after computers. Quantum computing will further that, introducing Q*. Galois theory will need to be modified, since set C is inconsistent.
Much is owned to Gordon Moore, who passed today at 94, and other pioneers.
Who "forces computers to us the sets R or C"? I see no attempt to do that with a digital computer. As far as I know, it is not even possible in any practical sense.
I understand (though am dismayed) by usage of the terms *real* and *complex* as type names when what is meant are variations of *float* and *double*. There is similar reason to avoid *integer* and I fancy *int* as more than an abbreviation. :).
For myself, I do not expect to see mathematics reduced to the computable. It doesn't bother me to leave analysis and calculus alone. It's not worth arguing about.
Another question about "Mathematics becomes an experimental science. As Voevodsky envisioned, computers can also flawlessly provide logical and mathematical proofs, without errors."
My impression is that computer-based proof systems are not yet that flawless and in particular are not able to proof all offered theorems without assistance. There is not yet, in particular, the provision of a polynomial-time prover for a problem that would have us answer P = NP? in the affirmative.
As we look to explain prime numbers objectively, it is important to realize that a prime number is not a mysterious accident in set Z, which happens to use base 10, or subjective.
A prime number is an objective value, valid for friends and foes.
This works because a number is a semiotic property.
The name can change, the referent can also change, but the sense stays constant: a prime number is a prime number.
In other words, a number is a 1:1 mapping between a name and a value.
Different pairs (name, value) have the same sense for any base.
As we can jokingly summarize, by this mathematical property, a prime number on Earth is a prime number in Betelgeuse. An objective property, even though numbers might be subjective.
A private message asks what is more special about quantum computing? Speed?
No. In the original meaning of the word "soviet", quantum computing shows that numbers are not a soviet thing.
Numbers obey a physical law. Massachusetts Institute of Technology Peter Shor was the first to say it, in 1994. So-called "pure" mathematics is, after all, governed by objective laws.
This we call a "wormhole", liking "pure" mathematics with physics. To the horror of fellow mathematicians who write to me, complaining. Read Ramanujan.
Humans are also linked. To the horror of die-hard soviets, who write from ... Ukraine. Go watch the online movie "The Embittered City" ... made before the war.
Understanding number in semiotics helps in using it, bringing objectivity to mathematics albeit preserving subjectivity. Those who believe in continuity ... go buy a good microscope.
Code benefits, and does not need anymore to follow illusions. R=Q --> B.
Soviet thinking loses.
The reference of a sentence is its truth value, whereas its sense is the thought that it expresses. Frege justified the distinction in a number of ways.
One can see a number as pointing to its truth value (called a value) and expressing a thought (called its digit in the base used). The digit is the sense, the meaning of the number.
The digit can change at will, subjectively, but the value stays objectively fixed.
Thus one can write F in hexadecimal, while the value is 15 in base 10. The number is this 1:1 mapping between a name ("F") and a value (15).
Or, in reverse, one can use 15 for the name, pointing to the value F.
Or, one can use 1111 in 64-bit signed binary for the name, pointing to the same value (15).
A number is seen as this flexible semiotic concept, very used in computers.
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