I believe that (mathematical) proofs aren’t easily reducible to axiomatic proofs, and that proofs have been, and still are, profoundly social by their nature, although I don’t know if that will continue indefinitely. I probably won’t find the time to write a large post on this topic that I’ve been thinking of, so I want to quote here one observation that’s been on my mind recently, in case someone finds it useful.
Eliezer quotes (in the post which I, with accordance to the above, see as wrong-headed in some respects) one definition of proof that he disagrees with: “A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing.”
I think this is too vague as a definition of proof and doesn’t really work, but it does capture the social/communication aspect of it I consider important. A thinks X is right and P is a proof of X. It isn’t enough that P convinces A that X is right; when A communicates P to B, it should convince them too.
A few days ago, I was rereading Vladimir Uspensky’s essay on the philosophy of mathematics that I read many years ago and forgot. In the section about proofs, Uspensky offers this informal definition:
A proof is a convincing argument that convinces us to such a degree that we can with its help convince others.
That is, it isn’t enough that B is convinced by P that X is right; P should be such that B should be able to spread the gospel onwards. A proof doesn’t just bridge a void between two minds; it’s capable of leaping on and on. It’s a virus with conviction as its payload: it instills conviction in its host and can spread itself (rather than merely conviction) to others.
I’ve been musing since then about this addition to what I’d thought of as an informal social definition of a proof. I’ve been going back and forth about how necessary and profound it is, but I think I’m converging on forth.
A proof is a convincing argument that convinces us to such a degree that we can with its help convince others.
I’m reminded of advice I’ve seen on when, in the martial arts, you may consider yourself a master: when people come to your students, asking “Please teach us.”
Although I disagree with the social concept of mathematical proof. There is something social going on, but there is also something that is not social, but is mathematical, existing independently of ourselves, outside space and time. The latter is the actual proof, without which a statement is not a theorem, and the former is people convincing other people that the latter exists and showing them where it is. Both of these things are called “proofs”, and both of them are necessary to the practice of mathematics. The social part depends on the mathematical part for its value, just as prospecting for oil has no value unless, in the end, there is oil where you say there is.
It sounds to me that you (or Uspensky) are trying to define “proof” to mean all good things. Perhaps this come from a belief that mathematics is synonymous with proof. Arguing about definitions is generally bad and one common failure mode is to define all good things together. Have you read Thurston? He found a trade-off between proofs (in the usual sense) and other goals.
It sounds to me that you (or Uspensky) are trying to define “proof” to mean all good things.
I don’t understand what that means, and what exactly do you think it’s wrong with the definition offered. Certainly it doesn’t encompass definitions, axioms, conjectures, intuition, background knowledge and other good things in mathematics.
I know and value Thurston’s paper, and again, don’t quite see the relevance.
I don’t understand how Uspensky’s definition is different from Eliezer’s. Is there some minimum number of people a proof has to convince? Does it have to convince everyone? If I’m the only person in the world, is writing a proof impossible, or trivial? It seems that both definitions are saying that a proof will be considered valid by those people who find it absolutely convincing. And those people who do not find it absolutely convincing will not consider it valid. More importantly, it seems that this is all those two definitions are saying, which is why neither of them is very helpful if we want something more concrete than the colloquial sense of proof.
As I understand Eliezer’s definition: “Your text is proof if it can convince me that the conclusion is true”
As I understand Uspenskiy’s definition: “Your text is proof if it can convince me that the conclusion is true and is I am willing to reuse this text to convince other people”.
The difference is whether the mere text convinces me that I myself can also use it succesfully. Of course this has to rely on social norms for convincing arguments in some way.
Disclosure: I have heard the second definition from Uspenskiy first-person, and I have never seen Eliezer in person.
Does Uspenskiy have an opinion on Zero-knowledge proofs? They differ from standard proofs in that they have a probability of being wrong (which can be as small as you want), and the key property which is that if I use one to convince you of something, you aren’t able to use it to convince anyone else.
Does anyone consider them the proofs in the ordinary sense?
I could ask him, but given that experience of verification of ZKP is an example personal/non-transferrable evidence, I see no question here.
And in some sense, ZKP proofs are usualy proofs of knowledge. If you represent ZK prover as a black box with secret information inside that uses random number generation log and communication log as sources to calculate its next message, access to this blackbox in such form is enough to extract some piece of data. This piece of data makes the proven statement trivial to verify or to prove conventionally—or even to play prover in ZKP. So all the efforts in ZKP are about showing you know some secret without letting me know the secret.
I believe that (mathematical) proofs aren’t easily reducible to axiomatic proofs, and that proofs have been, and still are, profoundly social by their nature, although I don’t know if that will continue indefinitely. I probably won’t find the time to write a large post on this topic that I’ve been thinking of, so I want to quote here one observation that’s been on my mind recently, in case someone finds it useful.
Eliezer quotes (in the post which I, with accordance to the above, see as wrong-headed in some respects) one definition of proof that he disagrees with: “A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing.”
I think this is too vague as a definition of proof and doesn’t really work, but it does capture the social/communication aspect of it I consider important. A thinks X is right and P is a proof of X. It isn’t enough that P convinces A that X is right; when A communicates P to B, it should convince them too.
A few days ago, I was rereading Vladimir Uspensky’s essay on the philosophy of mathematics that I read many years ago and forgot. In the section about proofs, Uspensky offers this informal definition:
That is, it isn’t enough that B is convinced by P that X is right; P should be such that B should be able to spread the gospel onwards. A proof doesn’t just bridge a void between two minds; it’s capable of leaping on and on. It’s a virus with conviction as its payload: it instills conviction in its host and can spread itself (rather than merely conviction) to others.
I’ve been musing since then about this addition to what I’d thought of as an informal social definition of a proof. I’ve been going back and forth about how necessary and profound it is, but I think I’m converging on forth.
I’m reminded of advice I’ve seen on when, in the martial arts, you may consider yourself a master: when people come to your students, asking “Please teach us.”
Although I disagree with the social concept of mathematical proof. There is something social going on, but there is also something that is not social, but is mathematical, existing independently of ourselves, outside space and time. The latter is the actual proof, without which a statement is not a theorem, and the former is people convincing other people that the latter exists and showing them where it is. Both of these things are called “proofs”, and both of them are necessary to the practice of mathematics. The social part depends on the mathematical part for its value, just as prospecting for oil has no value unless, in the end, there is oil where you say there is.
It sounds to me that you (or Uspensky) are trying to define “proof” to mean all good things. Perhaps this come from a belief that mathematics is synonymous with proof. Arguing about definitions is generally bad and one common failure mode is to define all good things together. Have you read Thurston? He found a trade-off between proofs (in the usual sense) and other goals.
I don’t understand what that means, and what exactly do you think it’s wrong with the definition offered. Certainly it doesn’t encompass definitions, axioms, conjectures, intuition, background knowledge and other good things in mathematics.
I know and value Thurston’s paper, and again, don’t quite see the relevance.
I don’t understand how Uspensky’s definition is different from Eliezer’s. Is there some minimum number of people a proof has to convince? Does it have to convince everyone? If I’m the only person in the world, is writing a proof impossible, or trivial? It seems that both definitions are saying that a proof will be considered valid by those people who find it absolutely convincing. And those people who do not find it absolutely convincing will not consider it valid. More importantly, it seems that this is all those two definitions are saying, which is why neither of them is very helpful if we want something more concrete than the colloquial sense of proof.
As I understand Eliezer’s definition: “Your text is proof if it can convince me that the conclusion is true”
As I understand Uspenskiy’s definition: “Your text is proof if it can convince me that the conclusion is true and is I am willing to reuse this text to convince other people”.
The difference is whether the mere text convinces me that I myself can also use it succesfully. Of course this has to rely on social norms for convincing arguments in some way.
Disclosure: I have heard the second definition from Uspenskiy first-person, and I have never seen Eliezer in person.
Does Uspenskiy have an opinion on Zero-knowledge proofs? They differ from standard proofs in that they have a probability of being wrong (which can be as small as you want), and the key property which is that if I use one to convince you of something, you aren’t able to use it to convince anyone else.
Does anyone consider them the proofs in the ordinary sense?
I could ask him, but given that experience of verification of ZKP is an example personal/non-transferrable evidence, I see no question here.
And in some sense, ZKP proofs are usualy proofs of knowledge. If you represent ZK prover as a black box with secret information inside that uses random number generation log and communication log as sources to calculate its next message, access to this blackbox in such form is enough to extract some piece of data. This piece of data makes the proven statement trivial to verify or to prove conventionally—or even to play prover in ZKP. So all the efforts in ZKP are about showing you know some secret without letting me know the secret.