Conjecture:fundamental mathematical group

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Discussion Overview

The discussion revolves around the concept of a fundamental mathematical group, with a focus on the role of non-negative integers and irrational numbers in mathematics. Participants explore the implications of defining mathematics from basic concepts and operations, while also addressing the philosophical aspects of mathematical evolution.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Philosophical

Main Points Raised

  • One participant proposes that a fundamental group consists of non-negative integers and irrational numbers, with addition and subtraction as operations and zero as the identity element.
  • Another participant challenges the exclusion of set theory, arguing that numbers can be derived from ZFC, and questions the fundamental nature of non-negative integers due to their lack of closure under subtraction.
  • A participant emphasizes the distinction between "natural" objects and "invented" objects in mathematics, suggesting that the evolution of mathematics should be considered in defining fundamental concepts.
  • Some participants argue that a proper group must include all integers to maintain closure under addition and subtraction, acknowledging that the initial proposal lacked this requirement.
  • There is a discussion about the historical context of natural numbers and the development of mathematical concepts, with references to Kronecker's views on the nature of integers and the evolution of negative numbers.
  • One participant critiques the approach of deriving mathematics solely from natural numbers, suggesting that modern mathematics relies heavily on proofs and formal systems like ZFC.

Areas of Agreement / Disagreement

Participants express differing views on the fundamental nature of non-negative integers versus integers, the role of set theory, and the historical development of mathematical concepts. There is no consensus on the validity of the proposed group or the foundational aspects of mathematics.

Contextual Notes

Participants highlight limitations in the initial proposal regarding the definition of a group and the closure properties of operations. The discussion also reflects varying interpretations of what constitutes "natural" versus "invented" mathematical objects.

  • #61
SW VandeCarr said:
You're not talking about a sieve, are you? To me, a sieve is a dumb algorithm which tests all odd numbers not ending in 5. That's not what I meant. I meant a smart algorithm which can produce all the primes and only the primes. Such an algorithm would also tell us the number of primes over any specified interval of natural numbers.

What is a "smart" algorithm? Do you mean an algorithm that has a polynomial complexity? I can write you an algorithm that can generate all primes (and only primes) and one for telling you how many primes are in a given finite subset of the naturals. You can tell the algorithm to output infinity whenever you enter a infinite subset.
 
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  • #62
Focus said:
What is a "smart" algorithm? Do you mean an algorithm that has a polynomial complexity? I can write you an algorithm that can generate all primes (and only primes) and one for telling you how many primes are in a given finite subset of the naturals. You can tell the algorithm to output infinity whenever you enter a infinite subset.
So if I enter the infinite subset { 2, 4, 6, ... }, it will return infinity?
 
  • #63
Focus said:
What is a "smart" algorithm? Do you mean an algorithm that has a polynomial complexity? I can write you an algorithm that can generate all primes (and only primes) and one for telling you how many primes are in a given finite subset of the naturals. You can tell the algorithm to output infinity whenever you enter a infinite subset.

Then why do we need to estimate the number of primes less than x asymptotically with x/ln(x)?
 
  • #64
SW VandeCarr said:
Then why do we need to estimate the number of primes less than x asymptotically with x/ln(x)?
Because the algorithm is time consuming. An algorithm must complete in a finite number of steps. You would be suprised to learn how large some numbers can be and still be considered finite.
 
  • #65
jimmysnyder said:
Because the algorithm is time consuming. An algorithm must complete in a finite number of steps. You would be suprised to learn how large some numbers can be and still be considered finite.

Of course. That's why de facto, we have no provable formula that will generate all the primes and only the primes.
 
  • #66
SW VandeCarr said:
Of course. That's why de facto, we have no provable formula that will generate all the primes and only the primes.
Algorithm. You have not proved that there is no algorithm, you have only proved that you think there isn't one.
 
  • #67
SW VandeCarr said:
Of course. That's why de facto, we have no provable formula that will generate all the primes and only the primes.
No, that's obviously not a proof that such a formula does not exist. We do have an algorithm which generates all primes (sieve of Eratosthenes). It is an algorithm, admittedly not very efficient, but perfectly valid still.

Can you prove to me that there is no polynomial (possibly of very large degree) such that P(n) is the n-th prime ? That would be another algorithm. It would suffice to evaluate the polynomial at every integer to get all the primes. It would still take an infinite time, but it would be more efficient than the sieve of Eratosthenes.
 
  • #68
humanino said:
No, that's obviously not a proof that such a formula does not exist. We do have an algorithm which generates all primes (sieve of Eratosthenes). It is an algorithm, admittedly not very efficient, but perfectly valid still.

Can you prove to me that there is no polynomial (possibly of very large degree) such that P(n) is the n-th prime ? That would be another algorithm. It would suffice to evaluate the polynomial at every integer to get all the primes. It would still take an infinite time, but it would be more efficient than the sieve of Eratosthenes.

No. I'm saying we don't have an efficient formula to generate the nth prime. I'm not saying one can't possibly exist.
 
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  • #69
jimmysnyder said:
Algorithm. You have not proved that there is no algorithm, you have only proved that you think there isn't one.

See my response to humanino. Also, mathematical formulas with numerical outputs and algorithms are implemented the same way, as computational steps.
 
  • #70
humanino said:
Can you prove to me that there is no polynomial (possibly of very large degree) such that P(n) is the n-th prime?
Yes. Let P be a polynomial of degree m such that P(n) is prime for all n.
P(x) = a_mx^m + ... + a_0
Then P(1) = p where p is a prime, so P(1) = 0 (mod p). So for any k,
P(1 + kp) = a_m(1+pk)^m + ... + a_0
= a_m + a_mb_m + ... + a_1 + a_1b_1 + a_0
(where b_i is divisible by p for all i)
= P(1) mod p
= 0 mod p
so P(1 + kp) = 0 (mod p) and either P(1 + kp) is divisable by p and is not prime, or is 0. But P only has m zeros or is itself the zero polynomial.
 
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  • #71
jimmysnyder said:
...
I never had any doubt that you would know. I suspect you may even be able to come up with another proof :smile:
Thanks for the answer.
 
  • #72
Locked pending moderation.
 

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