Conjecture:fundamental mathematical group

[SW VandeCarr]

Of course, I doubt Euler would have any difficulty understanding that set in various other applications. For example, I doubt Euler would have any difficulty understanding the hierarchy that set describes (depicted below as a graph), or its application as describing a container containing two containers, one empty, and the other containing an empty container.

<br /> \begin{matrix}<br /> &amp; &amp; \bullet \\<br /> &amp; \swarrow &amp; &amp; \searrow \\<br /> \bullet &amp; &amp; &amp; &amp; \bullet \\<br /> &amp; &amp; &amp; &amp; \downarrow \\<br /> &amp; &amp; &amp; &amp; \bullet<br /> \end{matrix}<br />

Euler probably invented graph theory with his Konigsberg Bridge problem and the Eulerian circuit.

As far as using a container model for understanding the empty set, one could argue that a container is something. The conceptual problem, as it see it, is that while the idea of a collection is intuitive, the idea of an empty container inside of a container that is otherwise empty appears to contradict the stricture that there is only one empty set.

And the research I recall is that our instinctive concept of counting number starts becoming fuzzy around the number 5, if not earlier. I believe there's even a pretty good case that our instinctive notion of quantity only has three categories: 0, 1, and more than one. (Although I suspect that one is the fault of language, not instinct)

Whatever instinctive notions pre-school children have of number, most are able to quickly learn how to generate longer integer sequences.

 

[Ravid]

Perhaps I could mount a serious argument if you actually explained why you think the rationals are derived but the irrationals not. Both arose as a "conscious extension of something more basic." Both have examples arising in nature, as I indicated. In post #23 you gave historical reasons for choosing the integers - fine. This fails to explain the irrationals, which as I'm sure you know, historically arose later than the rationals.

 

[SW VandeCarr]

Perhaps I could mount a serious argument if you actually explained why you think the rationals are derived but the irrationals not. Both arose as a "conscious extension of something more basic." Both have examples arising in nature, as I indicated. In post #23 you gave historical reasons for choosing the integers - fine. This fails to explain the irrationals, which as I'm sure you know, historically arose later than the rationals.

I don't know how to say it any better than I've already said it. Fractions are compositions of integers. The irrationals are not. No one consciously extended the concept of a fraction to the true nature of the irrationals. It took some time before people realized that you couldn't express pi with a fraction. 22/7 was used a lot by the ancients (and it's not a bad approximation), but they knew it wasn't quite right. Fractions were invented The irrationals were discovered. They did not arise from any conscious extension of fractions.They were discovered when people tried to express them as fractions. They were an unwelcome intrusion into an otherwise well ordered world.

 

[apeiron]

I'm simply asserting, based the historical record, that integers are fundamental without attempting to find the neurophysiologic basis for it. All I want to do is distill the record and make it a basis for teaching math. .

Sorry, I didn't realize your mission here was paedagogic. You posted in a philosophy forum.

Even so, that would seem only to make it more important to base teaching on the things brains find most easy to grasp.

 

[Ravid]

I don't know how to say it any better than I've already said it. Fractions are compositions of integers. The irrationals are not. No one consciously extended the concept of a fraction to the true nature of the irrationals. It took some time before people realized that you couldn't express pi with a fraction. 22/7 was used a lot by the ancients (and it's not a bad approximation), but they knew it wasn't quite right. Fractions were invented The irrationals were discovered. They did not arise from any conscious extension of fractions.They were discovered when people tried to express them as fractions. They were an unwelcome intrusion into an otherwise well ordered world.

The distinction you make here is arbitrary. \sqrt{2} is obtained from 2. The process needed to obtain it is different from that to obtain 1/2, but the difference is not fundamental. People found it hard to accept irrationals, but they were a conscious extension of the rational number counting system that allowed people to express ratios in Euclidean geometry that they otherwise couldn't, just as rationals were an extension of integers allowing people to express parts of a whole (which similarly arise as ratios in Euclidean geometry).

In a more modern sense: rationals are needed so that you can divide. Algebraic integers are needed so that you can take square roots. Trancendentals are needed so that you can take limits. They are all extensions of simpler systems, both historically and logically.

 

[SW VandeCarr]

The distinction you make here is arbitrary. \sqrt{2} is obtained from 2.

The distinction I make is that the rationals can be expressed exactly (as a ratio of integers) and the irrationals can only be approximated. If that's arbitrary, so be it.

 

[Ravid]

The distinction I make is that the rationals can be expressed exactly (as a ratio of integers) and the irrationals can only be approximated. If that's arbitrary, so be it.

\sqrt 2 is an exact expression of the square root of two. What I suggest you mean is that rationals can be expressed exactly as finite combinations of integers with finitary (binary) algebraic operations, e.g. division, as opposed to as solutions of an equation or a limit. But then again, \sqrt is an unary algebraic operation on \mathbb R^+, so perhaps not even that.

The point is that this difference alone is not sufficient to distinguish between invention or discovery. You may say that the rationals and irrationals were constructed (though admittedly by different methods) or that the constructions were simply a realisation of something that was already 'there'. You should be worried about whether the distinctions you make are arbitrary, because if they are you will find your position difficult to defend.

 

[SW VandeCarr]

\sqrt 2 is an exact expression of the square root of two. What I suggest you mean is that rationals can be expressed exactly as finite combinations of integers with finitary (binary) algebraic operations, e.g. division, as opposed to as solutions of an equation or a limit. But then again, \sqrt is an unary algebraic operation on \mathbb R^+, so perhaps not even that.

The point is that this difference alone is not sufficient to distinguish between invention or discovery. You may say that the rationals and irrationals were constructed (though admittedly by different methods) or that the constructions were simply a realisation of something that was already 'there'. You should be worried about whether the distinctions you make are arbitrary, because if they are you will find your position difficult to defend.

OK. Instead of all irrationals, suppose I said only the transcendental numbers were not derivative.

 

[SW VandeCarr]

Quoted: If an algorithm didn't generate all the primes and only the primes, I wouldn't have called it such.

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.

 

[Jimmy Snyder]

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.

That's not how the sieve works. It doesn't test any numbers at all and it deals with numbers divisible by 5 in the same way that it treats numbers divisible by any other prime. If it is used to find out if a particular number is prime it is an algorithm. It can't be considered an algorithm to find all prime numbers though since it would take an infinite number of steps to do that and an algorithm, by definition, can only take a finite number of steps.

 

[Focus]

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.

 

[Jimmy Snyder]

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?

 

[SW VandeCarr]

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)?

 

[Jimmy Snyder]

Then why do we need to estimate the number of primes less than x asymptotically with x/ln(x)?

Because the algoritm 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.

 

[SW VandeCarr]

Because the algoritm 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.

 

[Jimmy Snyder]

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.

 

[humanino]

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.

 

[SW VandeCarr]

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.

 

[SW VandeCarr]

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.

 

[Jimmy Snyder]

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.

 

[humanino]

...

I never had any doubt that you would know. I suspect you may even be able to come up with another proof
Thanks for the answer.

 

[Evo]

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