Old proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (some prime factor of (v_n + 1)).
"New" proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (v_n + 1).
Doesn't seem terribly different or more constructive or any such thing to me. I'd say both of these have the same fundamental content, just framed slightly differently.
[Just to head off what might superficially seem to be a significant distinction: Note that, in both proofs, we use the (constructive) fact that every integer > 1 has a prime factor, and even in the "new" proof, in order to actually extract an infinite stream of primes, one must actually carry out this process of finding prime factors for integers > 1 on demand.]
Surprised this got published. This theorem is literally day-1 number theory. And unless I'm missing something, this paper is just rephrasing it in a slightly different way.
That's not really an accurate representation of the old proof. You've turned it around to make it look like the new one, but that's not how it is originally laid out at all.
The fundamental problem (for constructivists) with Euclid's proof is that it assumes the existence of a set of all primes, and then proves that that set is incorrect (either contains a composite, or doesn't contain at least one specific prime).
Saidak's proof, on the other hand, only assumes the existence of a single integer, and then constructs infinitely many primes from that integer.
Your restatement of Euclid's proof is basically "look, Euclid and Saidak proved the same thing", not "look, Euclid's proof is constructive too".
I think the point is you can easily remove the "deficiencies" of Euclid's proof. Basically, say: let p_1, ..., p_n be the first n primes. We can construct a new prime by examining p_1...p_n + 1. This process can be continued indefinitely.
In fact, I think the statement "this process can be continued indefinitely" is quite misleading -- to formalize this, I think you need proof by contradiction. So, the difference is the syntax used, not the actual semantics of the proof.
There a ways to formulate "the infinity of the prime numbers" without any recourse to a previously-constructed set of all prime numbers (or even all natural numbers). Two such ways are:
1. "For any natural number n, there is a prime number greater than n."
2. "For any natural number m, and for any list of m prime numbers p_1, ..., p_m, there exists a prime number not on that list."
Both statements have a perfectly fine constructive proof. The second statement is more or less precisely what Euclid proved. The first statement follows from the second one with a bit more work.
A) Euclid never assumes the existence of a finite set of all primes (doing so at the outset is a framing adopted only by later mathematicians); people just frequently erroneously think Euclid's proof did this, never having actually read it.
B) Regardless, I don't particularly care how Euclid phrased his proof; I never mentioned Euclid. Certainly, mathematicians have been aware that Euclid's insight could be framed in the way I called an "old proof" for a long, long time. The distinctions between the various framings here are trivial.
The idea of using `consecutive numbers are coprime` as the sole property for this concise proof is quite remarkable.
I would definitely be interested to know if there are any other such simple proofs exist for other theorems. (i.e. ones where a new proof simplifies it massively by using a simpler property).
The proof just given is conceptually even simpler than the
original proof due to Euclid, since it does not use Eudoxus’s method of“reductio ad absurdum,” proof by contradiction. And unlike most other proofs of the theorem, it does not require Proposition 30 of Elements (sometimes called “Euclid’s Lemma”) that states: if p is a prime and p|ab, then either p|a or p|b. Moreover, our proof is constructive, and it gives integers with an arbitrary number of prime factors.
Edit: Actually, even though the article seems to imply that the classic proof ("most proofs") uses prop 30, it doesn't really seem to.
The article makes suggestions here about the classic proof which aren't true.
Euclid's proof of the infinitude of the primes was not phrased in terms of an overarching reductio ad absurdum (and even had it counterfactually been, mathematicians would've long ago been able to trivially rephrase it so as not to be, showing "For any finite set of primes, there is some further prime" directly).
And the classic proof of the infinitude of the primes does not anywhere use Proposition 30 of the Elements (see for yourself at http://aleph0.clarku.edu/~djoyce/elements/bookIX/propIX20.ht... ; Proposition 31 (that every composite has some prime factor) is used, but this in turn is argued for without any invocation of Proposition 30).
Where in the classic proof would you imagine "if p is a prime and p | ab, then p | a or p | b" would come up?
It seems to me that the original proof also uses "consecutive numbers are coprime" as the sole property. In particular, it looks at P = p_1 * ... * p_n and says that it is coprime to Q = P + 1; hence, there must exist another prime number.
The proof isn't referring to n, n+1, and n+2 -- it refers to n, n+1, and n(n+1) + 1. In your example, this would be 2, 3, and 2*3 + 1 = 7, which in fact include 3 prime factors.
You've misread the proof. They do not argue that n, n + 1, and n + 2 are co-prime; they only note that n, (n + 1), and n * (n + 1) + 1 are co-prime [and more generally they continue in the same fashion, using the basic fact that n and n + 1 are co-prime for any n].
The proof is sound. However, it is also not particularly different in essence from the classic proof; considering it so is misguided (see https://news.ycombinator.com/item?id=10889290).
This discussion illustrates an interesting point -- it can be a little tricky to judge whether two proofs are actually "different" or not.
When I was a grad student I interrupted a study session to raise the simple question -- can a theorem really have two different proofs? We all thought the answer was yes, but the discussion went on for several minutes until I settled it by contriving a stupidly simple and synthetic set of axioms to construct a example.
Basically, the axioms were:
"All As are also Bs"
"All As are also Cs"
"All Bs are also Ds"
"All Cs are also Ds"
Yes; putting essentially the same example another way, we might feel intuitively that there are two separate proofs of the propositional theorem that "B AND C" implies "B OR C".
Edit: Or, following up on danharaj's framing, we might feel intuitively that there are two separate proofs of "The set {B, C} is inhabited".
It can get more complicated. When your logic can talk about equivalence of equivalences of proofs, then you can have two proofs that are equivalent in two ways that are not equivalent. Or, you could have two proofs that are equivalent in two ways whose equivalence can be proved, but maybe also in more than one way. You can iterate equivalences of equivalences in such a logic and equality becomes a significantly richer concept.
The study of such mathematical structures is the study of what is called an (infinity,1)-topos.
For readers who aren't already familiar with this, I'll note that this is in large part what Homotopy Type Theory is about making easy to work with. http://homotopytypetheory.org/book/ is a good introduction.
One could argue that the proof is just that all As are Xs where all Xs are Ds and therefore all As are Ds. This factors out the detail of choosing one path over the other. You have of course still to show that such an X exists and can use either pair of the axioms establishing this but as far as this proof is concerned both axiom pairs are really equivalent. So are this really two different proofs? How would one define equivalence between proofs to begin with? Isomorphic graphs of applied derivation rules?
From the point of view of category-theoretic logic – which is the area I did my PhD in – there is a sense in which all classical proofs are equivalent.
Roughly speaking, the argument goes like this. Intuitionistic propositional logic can be modelled by cartesian closed categories. (The canonical reference for this correspondence is the book “Introduction to Higher Order Categorical Logic” by Lambek and Scott.) But a lemma due to André Joyal shows that, if a cartesian closed category is a model of classical logic (i.e. validates the law of excluded middle), then there is at most one proof of any given implication: in the jargon, the category is a poset.
I'd like to draw attention to a fact already pointed out by Chinjut in this thread: The original formulation of Euclid's proof is in fact entirely constructive, contrary to what the article claims.
Michael Hardy and Catherine Woodgold have written a very nice and readable account on the misconceptions about Euclid's proof (published in 2009). I'm sorry that I can only give a paywalled link; at least the first two pages are available. http://link.springer.com/article/10.1007%2Fs00283-009-9064-8
Incidentally, with the specific situation at hand, the question "constructive vs. nonconstructive" is slightly moot. This is because there is a certain metatheorem in mathematical logic which states: If there is a nonconstructive proof of a statement, then there is also a constructive proof.
Of course this metatheorem doesn't apply to arbitrary statements, only to statements of a specific logical form (so called "geometric sequents"). But the statement "there are infinitely many prime numbers" can be put into such a form. Also, in case you are wondering, this metatheorem admits itself a constructive proof.
Summarizing, there is a mechanical way to turn any nonconstructive proof of the infinitude of the primes into a constructive one.
The key words to look up here are "double-negation translation" and "Friedman's trick". Fantastically, the double-negation translation turns out to be "the same as" the continuation-passing style transformation, if viewed from the right angle. Some pointers are in this slide deck: http://rawgit.com/iblech/talk-constructive-mathematics/maste...
I wonder, do proofs of this kind, where some process is repeated forever to prove the thesis, have an infinite length? Could this be made into a finite number of steps in some formal system? Using a rule of induction maybe?
33 comments
[ 3.3 ms ] story [ 83.9 ms ] threadp<span class="_ _3"></span>ositiv<span class="_ _1"></span>e
"New" proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (v_n + 1).
Doesn't seem terribly different or more constructive or any such thing to me. I'd say both of these have the same fundamental content, just framed slightly differently.
[Just to head off what might superficially seem to be a significant distinction: Note that, in both proofs, we use the (constructive) fact that every integer > 1 has a prime factor, and even in the "new" proof, in order to actually extract an infinite stream of primes, one must actually carry out this process of finding prime factors for integers > 1 on demand.]
http://wstein.org/ent/ent.pdf
Also seems like a pretty baseless claim to say that this proof has never been done before.
The fundamental problem (for constructivists) with Euclid's proof is that it assumes the existence of a set of all primes, and then proves that that set is incorrect (either contains a composite, or doesn't contain at least one specific prime).
Saidak's proof, on the other hand, only assumes the existence of a single integer, and then constructs infinitely many primes from that integer.
Your restatement of Euclid's proof is basically "look, Euclid and Saidak proved the same thing", not "look, Euclid's proof is constructive too".
In fact, I think the statement "this process can be continued indefinitely" is quite misleading -- to formalize this, I think you need proof by contradiction. So, the difference is the syntax used, not the actual semantics of the proof.
1. "For any natural number n, there is a prime number greater than n."
2. "For any natural number m, and for any list of m prime numbers p_1, ..., p_m, there exists a prime number not on that list."
Both statements have a perfectly fine constructive proof. The second statement is more or less precisely what Euclid proved. The first statement follows from the second one with a bit more work.
A) Euclid never assumes the existence of a finite set of all primes (doing so at the outset is a framing adopted only by later mathematicians); people just frequently erroneously think Euclid's proof did this, never having actually read it.
Rather, Euclid constructively frames things, directly arguing that, for any finite set of primes, there is furthermore some prime outside that set. See http://aleph0.clarku.edu/~djoyce/elements/bookIX/propIX20.ht....
B) Regardless, I don't particularly care how Euclid phrased his proof; I never mentioned Euclid. Certainly, mathematicians have been aware that Euclid's insight could be framed in the way I called an "old proof" for a long, long time. The distinctions between the various framings here are trivial.
I would definitely be interested to know if there are any other such simple proofs exist for other theorems. (i.e. ones where a new proof simplifies it massively by using a simpler property).
The proof just given is conceptually even simpler than the original proof due to Euclid, since it does not use Eudoxus’s method of“reductio ad absurdum,” proof by contradiction. And unlike most other proofs of the theorem, it does not require Proposition 30 of Elements (sometimes called “Euclid’s Lemma”) that states: if p is a prime and p|ab, then either p|a or p|b. Moreover, our proof is constructive, and it gives integers with an arbitrary number of prime factors.
Edit: Actually, even though the article seems to imply that the classic proof ("most proofs") uses prop 30, it doesn't really seem to.
Euclid's proof of the infinitude of the primes was not phrased in terms of an overarching reductio ad absurdum (and even had it counterfactually been, mathematicians would've long ago been able to trivially rephrase it so as not to be, showing "For any finite set of primes, there is some further prime" directly).
And the classic proof of the infinitude of the primes does not anywhere use Proposition 30 of the Elements (see for yourself at http://aleph0.clarku.edu/~djoyce/elements/bookIX/propIX20.ht... ; Proposition 31 (that every composite has some prime factor) is used, but this in turn is argued for without any invocation of Proposition 30).
Where in the classic proof would you imagine "if p is a prime and p | ab, then p | a or p | b" would come up?
The proof is sound. However, it is also not particularly different in essence from the classic proof; considering it so is misguided (see https://news.ycombinator.com/item?id=10889290).
When I was a grad student I interrupted a study session to raise the simple question -- can a theorem really have two different proofs? We all thought the answer was yes, but the discussion went on for several minutes until I settled it by contriving a stupidly simple and synthetic set of axioms to construct a example.
Basically, the axioms were:
"All As are also Bs" "All As are also Cs" "All Bs are also Ds" "All Cs are also Ds"
and the theorem was
"All As are also Ds"
Theorem: There exists an even natural number.
Proof 1:
Proof 2:Edit: Or, following up on danharaj's framing, we might feel intuitively that there are two separate proofs of "The set {B, C} is inhabited".
Very cool example!
The study of such mathematical structures is the study of what is called an (infinity,1)-topos.
Roughly speaking, the argument goes like this. Intuitionistic propositional logic can be modelled by cartesian closed categories. (The canonical reference for this correspondence is the book “Introduction to Higher Order Categorical Logic” by Lambek and Scott.) But a lemma due to André Joyal shows that, if a cartesian closed category is a model of classical logic (i.e. validates the law of excluded middle), then there is at most one proof of any given implication: in the jargon, the category is a poset.
See http://mathoverflow.net/a/43285/8217 for a proof.
I guess like how some sentences are provable with or without the axiom of choice.
They already did. Which of their four axioms do you think follows from the rest?
Michael Hardy and Catherine Woodgold have written a very nice and readable account on the misconceptions about Euclid's proof (published in 2009). I'm sorry that I can only give a paywalled link; at least the first two pages are available. http://link.springer.com/article/10.1007%2Fs00283-009-9064-8
Incidentally, with the specific situation at hand, the question "constructive vs. nonconstructive" is slightly moot. This is because there is a certain metatheorem in mathematical logic which states: If there is a nonconstructive proof of a statement, then there is also a constructive proof.
Of course this metatheorem doesn't apply to arbitrary statements, only to statements of a specific logical form (so called "geometric sequents"). But the statement "there are infinitely many prime numbers" can be put into such a form. Also, in case you are wondering, this metatheorem admits itself a constructive proof.
Summarizing, there is a mechanical way to turn any nonconstructive proof of the infinitude of the primes into a constructive one.
The key words to look up here are "double-negation translation" and "Friedman's trick". Fantastically, the double-negation translation turns out to be "the same as" the continuation-passing style transformation, if viewed from the right angle. Some pointers are in this slide deck: http://rawgit.com/iblech/talk-constructive-mathematics/maste...