You could equally say that everyone who looks at the rules of chess sees the same thing. In order to show some inevitability to ZFC, you have to show that unconnected parties arriving at it independently.
You could equally say that everyone who looks at the rules of chess sees the same thing. In order to show some inevitability to ZFC, you have to show that unconnected parties arriving at it independently.
On the one hand, why? I’m quite happy to say that chess exists. Not everyone will ever see chess, but not everyone will ever see Neptune. Among all the games that could be played, chess is but one grain of sand on the beach. But the grain of sand exists regardless of whether anyone sees it.
On the other hand, there has been, I believe, a substantial tendency for people devising alternative axioms for the concepts of sets to come up with things equiconsistent to ZFC or to subsets of ZFC, and with fairly direct translations between them. Compare also the concept of computability, where there is a very strong tendency for different ways to answer the question “what is computation?” to come up with equivalent definitions.
It is (I think) true that if you try to come up with an alternative foundation for mathematics you are likely to get something that’s equivalent to some subset of ZFC perhaps augmented with some kind of large cardinal axiom. But that doesn’t mean that ZFC is inevitable, it means that if you construct two theories both intended to “support” all of mathematics without too much extravagance, you can often more or less implement one inside the other.
But that doesn’t mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes’s book “Elementary set theory with a universal set”) which I’ll call NFUIC henceforward. This is consistent relative to ZFC, and indeed relative to something rather weaker than ZFC, and NFUIC + “all Cantorian sets are strongly Cantorian” (never mind exactly what that means) is equiconsistent with ZFC + some reasonably plausible large-cardinal axioms. OK, fine, so there’s a sense in which NFUIC is ZFC-like, at least as regards consistency strength. But NFUIC’s sets are most definitely not the same as ZFC’s sets. NFUIC has a universal set and ZFC doesn’t; ZFC’s sets are the same sizes as their sets-of-singletons and NFUIC’s often aren’t; NFU has lots and lots of urelements and ZFC has just the single empty set; etc. NFUIC is very unlike ZFC despite these relationships in terms of consistency strength.
[EDITED to add:] Here’s an analogy. You get the same computable functions whether you start with (1) Turing machines, (2) register machines, (3) lambda calculus, or (4) Post production systems. But those are still four very different foundations for computing, they suggest quite different possible hardware realizations and different kinds of notation, they have quite different performance characteristics, etc. (The execution times are admittedly all bounded by polynomials in one another. We could add (5) quantum Turing machines, in which case that would no longer be known to be true.) It’s very interesting that these all turn out to be equivalent in power in some sense, but I wouldn’t call that convergence or suggest that it tells us that (e.g.) lambda-calculus terms have any sort of more exalted metaphysical status than they would if it weren’t for that equivalence.
But that doesn’t mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes’s book “Elementary set theory with a universal set”) which I’ll call NFUIC henceforward.
Yes, ZFC is not quite such an isolated landmark of thinginess as computability is, which is why I said “a strong tendency”. And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infintiely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s system NF was the only significant system of set theory that was not known to be equiconsistent with ZFC,
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
But that doesn’t mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes’s book “Elementary set theory with a universal set”) which I’ll call NFUIC henceforward.
Yes, ZFC may be not quite such a starkly isolated landmark of thinginess as computability is, which is why I said “a strong tendency”. And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infinitely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s NF was the only significant system of set theory for which this had not been done. I don’t know if progress has been made on that since.
(ETA: I found this review of NF from 2011. Its consistency was still open then.)
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Randall Holmes says he has a proof of the consistency of NF relative to ZFC (and in fact something weaker, I think). He’s said this for a while, he’s published a few versions of his proof (mostly different in presentation in the interests of clarity, rather than patching bugs), and I think the general feeling is that he probably does have a proof but it hasn’t yet been thoroughly checked by others. (Who may be holding off because he’s still changing his mind about the best way of writing it down.)
The question is whether the rules of chess have mind-indepedent existence.
Among all the games that could be played, chess is but one grain of sand on the beach. But the grain of sand exists regardless of whether anyone sees it.
Where are these grains, ie the rules of every possible game? Are they in our universe, or some heavenly library of babel?
On the other hand, there has been, I believe, a substantial tendency for people devising alternative axioms for the concepts of sets to come up with things equiconsistent to ZFC or to subsets of ZFC, and with fairly direct translations between them. Compare also the concept of computability, where there is a very strong tendency for different ways to answer the question “what is computation?” to come up with equivalent definitions.
so how do we cash out the idea that these things are converging on an abstract object , rather than just converging? One way is put forward the counterfactual that if the abstract object were different, then the convergence would occur differently. But that seems rather against the spirit of what you are aiming.
Where are these grains, ie the rules of every possible game? Are they in our universe, or some heavenly library of babel?
Ask Max Tegmark. :)
I don’t believe in his Level IV multiverse, though. That is, I do draw a distinction between physical and abstract objects.
If you don’t know what your theory is, why undertake to defend it?
so how do we cash out the idea that these things are converging on an abstract object , rather than just converging?
That they are converging is enough. To quote an old saw variously attributed, in mathematics existence is freedom from contradiction.
Vague again. Realists think mathematical existence, is a real, literal existence for which non-contradiuction is a criterion, whereas antirealists think mathematical existence is a mere metaphor with no more content than non-contradiction.
Perhaps I should finish with my usual comment that studying philosophy is useful because it allows you to articulate your theories, or, failing that, to notice when there are no clear concepts behind your words.
Vague again. Realists think mathematical existence, is a real, literal existence for which non-contradiuction is a criterion, whereas antirealists think mathematical existence is a mere metaphor with no more content than non-contradiction.
Vague and empty slogans that might be picked over endlessly, and have been, to no useful purpose. Don’t bother expanding them; it’s a dead end.
Perhaps I should finish with my usual comment that studying philosophy is useful because it allows you to articulate your theories, or, failing that, to notice when there are no clear concepts behind your words.
Much of the source material has failed to learn that lesson, and is useful in this regard primarily as negative examples. Some philosophers even say so. As a discipline for forcing you to discover the fallibility of subjectively clear and distinct ideas, and to arrive at actually clear ideas that demonstrably work, there are better fields of study, for those who are able to learn. Software design, for example.
Before, you seemed to be pitching in with an opinion on anti-realism versus realism, now you seem to be sayign the whole debate is meaningless. Which is your true belief? It isn’t clear.
Much of the source material has failed to learn that lesson, and is useful in this regard primarily as negative examples. Some philosophers even say so. As a discipline for forcing you to discover the fallibility of subjectively clear and distinct ideas, and to arrive at actually clear ideas that demonstrably work, there are better fields of study, for those who are able to learn. Software design, for example.
Says who? As a software engineer, and philosopher (and scientist), I have found philosophy to be the best training for expressing abstract ideas clearly.
Are you a software engineer? Do you believe software engineering has taught you to be clear?
Lisp is worth learning for … the profound enlightenment experience you will have when you finally get it. That experience will make you a better programmer for the rest of your days, even if you never actually use Lisp itself a lot.
As a software engineer, and philosopher (and scientist), I have found philosophy to be the best training for expressing abstract ideas clearly.
Well, I have not. Which philosophers would you particularly recommend for this purpose? What in philosophy will assist our most gifted fellow humans in thinking previously impossible thoughts, or is worth learning for the profound enlightenment experience I will have when I finally get it?
Are you a software engineer?
Software design and implementation has been a large part of my job and my recreations for all of my adult years. I have never taken a course of study in the subject.
Do you believe software engineering has taught you to be clear?
For example, I was familiar with the fallacy of suggestively named tokens long before reading Eliezer wrote of it on LessWrong, the fallacy of taking the subjective feeling that a task is simple for its actual simplicity, and probably various other things that are now just a part of my mental furniture.
While the lessons are there to be learned, that does not mean that everyone will learn them. I have rolled my eyes many times over what I would call junk XML languages, where the creators have done no more than write down English names for every concept they can think of in some domain of discourse, sprinkle pointy brackets over them, write a DTD, and believe they’ve achieved something. They have not. In the field of procedural humanoid animation, in which I have worked, there have been many attempts to generate animation from a human-written script specifying the movements, but, well, it would take too long to say what I think is wrong with most of them that my own efforts do better. I once heard a distinguished researcher in the field even say “I am not interested in stupid implementation”, as if the real work was in thinking up a structure and the “stupid implementation” could be left to a few graduate students.
And are either Dijkstra or ESR in a position to directly compare the efficacy of software engineering as a means of clearly expressing philosophical ideas to the efficacy of philosophy as a means of clearly expressing philosophical ideas..ie, do they know anything about philosophy?
It’s not news that when two or more STEM type are gathered together they will recite the Mantra Against the Philosophers, in the expectation of reaping agreement and maybe even applause. It’s just not very significant.
It’s also not news that software engineering teaches you to express software engineering concepts clearly, and it’s not very relevant since the topic is expressing philosophical concepts. Pointing out that someone else is unclear doesn’t make you clear. Pointing about that you can be clear about X doesn’t make you clear about Y.
Which philosophers would you particularly recommend for this purpose?
Getting into conversations where there is mutual commitment to clear communication is the best practice, because you get instant feedback Learning the jargon of philosophy—there are a number of dictionary-style works—is also helpful: after all the jargon is tailored to just the kind of topic you were discussing.
What in philosophy will assist our most gifted fellow humans in thinking previously impossible thought
That’s a different topic, but it happens...philosophers have been criticised for entertaining ideas that are too weird, among other things.
or is worth learning for the profound enlightenment experience I will have when I finally get it?
That’s a different topic, but it happens...philosophers have been criticised for entertaining ideas that are too weird, among other things.
Weirdness is not the same thing as previously impossible thought. In the strongest form of “impossible thought” you will not even understand the claim being made enough that it registers with you.
It’s not news that when two or more STEM type are gathered together they will recite the Mantra Against the Philosophers, in the expectation of reaping agreement and maybe even applause.
I’m not sure it makes sense to label people like E.S. Raymond who are proclaim hacker values as STEM types. Raymond isn’t following the popular science narrative of logical positivism that you find with the typical STEM person.
Weirdness is not the same thing as previously impossible thought. In the strongest form of “impossible thought” you will not even understand the claim being made enough that it registers with you.
Whatever. It’s about the third or fourth change of topic.
Lisp is worth learning for … the profound enlightenment experience you will have when you finally get it. That experience will make you a better programmer for the rest of your days, even if you never actually use Lisp itself a lot.
Software design and implementation has been a large part of my job and my recreations for all of my adult years. I have never taken a course of study in the subject.
Do you believe software engineering has taught you to be clear?
It has certainly greatly influenced me in that regard. For example, I was familiar with the fallacy of suggestively named tokens long before reading Eliezer wrote of it on LessWrong, the fallacy of taking the subjective feeling that a task is simple for its actual simplicity, and probably various other things that are now just a part of my mental furniture.
While the lessons are there to be learned, that does not mean that everyone will learn them. I have rolled my eyes many times over what I would call junk XML languages, where the creators have done no more than write down English names for every concept they can think of in some domain of discourse, sprinkle pointy brackets over them, write a DTD, and believe they’ve achieved something. They have not. In the field of procedural humanoid animation, in which I have worked, there have been many attempts to generate animation from a human-written script specifying the movements, but, well, it would take too long to say what I think is wrong with most of them that my own efforts get right. I once heard a distinguished researcher in the field even say “I am not interested in stupid implementation”, as if she could just think up a structure and leave it to a few graduate students to implement.
You could equally say that everyone who looks at the rules of chess sees the same thing. In order to show some inevitability to ZFC, you have to show that unconnected parties arriving at it independently.
On the one hand, why? I’m quite happy to say that chess exists. Not everyone will ever see chess, but not everyone will ever see Neptune. Among all the games that could be played, chess is but one grain of sand on the beach. But the grain of sand exists regardless of whether anyone sees it.
On the other hand, there has been, I believe, a substantial tendency for people devising alternative axioms for the concepts of sets to come up with things equiconsistent to ZFC or to subsets of ZFC, and with fairly direct translations between them. Compare also the concept of computability, where there is a very strong tendency for different ways to answer the question “what is computation?” to come up with equivalent definitions.
It is (I think) true that if you try to come up with an alternative foundation for mathematics you are likely to get something that’s equivalent to some subset of ZFC perhaps augmented with some kind of large cardinal axiom. But that doesn’t mean that ZFC is inevitable, it means that if you construct two theories both intended to “support” all of mathematics without too much extravagance, you can often more or less implement one inside the other.
But that doesn’t mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes’s book “Elementary set theory with a universal set”) which I’ll call NFUIC henceforward. This is consistent relative to ZFC, and indeed relative to something rather weaker than ZFC, and NFUIC + “all Cantorian sets are strongly Cantorian” (never mind exactly what that means) is equiconsistent with ZFC + some reasonably plausible large-cardinal axioms. OK, fine, so there’s a sense in which NFUIC is ZFC-like, at least as regards consistency strength. But NFUIC’s sets are most definitely not the same as ZFC’s sets. NFUIC has a universal set and ZFC doesn’t; ZFC’s sets are the same sizes as their sets-of-singletons and NFUIC’s often aren’t; NFU has lots and lots of urelements and ZFC has just the single empty set; etc. NFUIC is very unlike ZFC despite these relationships in terms of consistency strength.
[EDITED to add:] Here’s an analogy. You get the same computable functions whether you start with (1) Turing machines, (2) register machines, (3) lambda calculus, or (4) Post production systems. But those are still four very different foundations for computing, they suggest quite different possible hardware realizations and different kinds of notation, they have quite different performance characteristics, etc. (The execution times are admittedly all bounded by polynomials in one another. We could add (5) quantum Turing machines, in which case that would no longer be known to be true.) It’s very interesting that these all turn out to be equivalent in power in some sense, but I wouldn’t call that convergence or suggest that it tells us that (e.g.) lambda-calculus terms have any sort of more exalted metaphysical status than they would if it weren’t for that equivalence.
Yes, ZFC is not quite such an isolated landmark of thinginess as computability is, which is why I said “a strong tendency”. And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infintiely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s system NF was the only significant system of set theory that was not known to be equiconsistent with ZFC,
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Yes, ZFC may be not quite such a starkly isolated landmark of thinginess as computability is, which is why I said “a strong tendency”. And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infinitely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s NF was the only significant system of set theory for which this had not been done. I don’t know if progress has been made on that since.
(ETA: I found this review of NF from 2011. Its consistency was still open then.)
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Randall Holmes says he has a proof of the consistency of NF relative to ZFC (and in fact something weaker, I think). He’s said this for a while, he’s published a few versions of his proof (mostly different in presentation in the interests of clarity, rather than patching bugs), and I think the general feeling is that he probably does have a proof but it hasn’t yet been thoroughly checked by others. (Who may be holding off because he’s still changing his mind about the best way of writing it down.)
The question is whether the rules of chess have mind-indepedent existence.
Where are these grains, ie the rules of every possible game? Are they in our universe, or some heavenly library of babel?
so how do we cash out the idea that these things are converging on an abstract object , rather than just converging? One way is put forward the counterfactual that if the abstract object were different, then the convergence would occur differently. But that seems rather against the spirit of what you are aiming.
Ask Max Tegmark. :)
I don’t believe in his Level IV multiverse, though. That is, I do draw a distinction between physical and abstract objects.
That they are converging is enough. To quote an old saw variously attributed, in mathematics existence is freedom from contradiction.
If you don’t know what your theory is, why undertake to defend it?
Vague again. Realists think mathematical existence, is a real, literal existence for which non-contradiuction is a criterion, whereas antirealists think mathematical existence is a mere metaphor with no more content than non-contradiction.
Perhaps I should finish with my usual comment that studying philosophy is useful because it allows you to articulate your theories, or, failing that, to notice when there are no clear concepts behind your words.
Vague and empty slogans that might be picked over endlessly, and have been, to no useful purpose. Don’t bother expanding them; it’s a dead end.
Much of the source material has failed to learn that lesson, and is useful in this regard primarily as negative examples. Some philosophers even say so. As a discipline for forcing you to discover the fallibility of subjectively clear and distinct ideas, and to arrive at actually clear ideas that demonstrably work, there are better fields of study, for those who are able to learn. Software design, for example.
Before, you seemed to be pitching in with an opinion on anti-realism versus realism, now you seem to be sayign the whole debate is meaningless. Which is your true belief? It isn’t clear.
Says who? As a software engineer, and philosopher (and scientist), I have found philosophy to be the best training for expressing abstract ideas clearly.
Are you a software engineer? Do you believe software engineering has taught you to be clear?
These two, for example, from opposite ends of the academic/professional spectrum:
E.W. Dijkstra, “The Humble Programmer”
E.S. Raymond, “How To Become A Hacker”
Well, I have not. Which philosophers would you particularly recommend for this purpose? What in philosophy will assist our most gifted fellow humans in thinking previously impossible thoughts, or is worth learning for the profound enlightenment experience I will have when I finally get it?
Software design and implementation has been a large part of my job and my recreations for all of my adult years. I have never taken a course of study in the subject.
For example, I was familiar with the fallacy of suggestively named tokens long before reading Eliezer wrote of it on LessWrong, the fallacy of taking the subjective feeling that a task is simple for its actual simplicity, and probably various other things that are now just a part of my mental furniture.
While the lessons are there to be learned, that does not mean that everyone will learn them. I have rolled my eyes many times over what I would call junk XML languages, where the creators have done no more than write down English names for every concept they can think of in some domain of discourse, sprinkle pointy brackets over them, write a DTD, and believe they’ve achieved something. They have not. In the field of procedural humanoid animation, in which I have worked, there have been many attempts to generate animation from a human-written script specifying the movements, but, well, it would take too long to say what I think is wrong with most of them that my own efforts do better. I once heard a distinguished researcher in the field even say “I am not interested in stupid implementation”, as if the real work was in thinking up a structure and the “stupid implementation” could be left to a few graduate students.
And are either Dijkstra or ESR in a position to directly compare the efficacy of software engineering as a means of clearly expressing philosophical ideas to the efficacy of philosophy as a means of clearly expressing philosophical ideas..ie, do they know anything about philosophy?
It’s not news that when two or more STEM type are gathered together they will recite the Mantra Against the Philosophers, in the expectation of reaping agreement and maybe even applause. It’s just not very significant.
It’s also not news that software engineering teaches you to express software engineering concepts clearly, and it’s not very relevant since the topic is expressing philosophical concepts. Pointing out that someone else is unclear doesn’t make you clear. Pointing about that you can be clear about X doesn’t make you clear about Y.
Getting into conversations where there is mutual commitment to clear communication is the best practice, because you get instant feedback Learning the jargon of philosophy—there are a number of dictionary-style works—is also helpful: after all the jargon is tailored to just the kind of topic you were discussing.
That’s a different topic, but it happens...philosophers have been criticised for entertaining ideas that are too weird, among other things.
That’s a different topic again.
Weirdness is not the same thing as previously impossible thought. In the strongest form of “impossible thought” you will not even understand the claim being made enough that it registers with you.
I’m not sure it makes sense to label people like E.S. Raymond who are proclaim hacker values as STEM types. Raymond isn’t following the popular science narrative of logical positivism that you find with the typical STEM person.
Whatever. It’s about the third or fourth change of topic.
Well, I have not. Which philosophers would you particularly recommend?
These two, for example, from opposite ends of the academic/professional spectrum:
E.W. Dijkstra, “The Humble Programmer”
E.S. Raymond, “How To Become A Hacker”
Software design and implementation has been a large part of my job and my recreations for all of my adult years. I have never taken a course of study in the subject.
It has certainly greatly influenced me in that regard. For example, I was familiar with the fallacy of suggestively named tokens long before reading Eliezer wrote of it on LessWrong, the fallacy of taking the subjective feeling that a task is simple for its actual simplicity, and probably various other things that are now just a part of my mental furniture.
While the lessons are there to be learned, that does not mean that everyone will learn them. I have rolled my eyes many times over what I would call junk XML languages, where the creators have done no more than write down English names for every concept they can think of in some domain of discourse, sprinkle pointy brackets over them, write a DTD, and believe they’ve achieved something. They have not. In the field of procedural humanoid animation, in which I have worked, there have been many attempts to generate animation from a human-written script specifying the movements, but, well, it would take too long to say what I think is wrong with most of them that my own efforts get right. I once heard a distinguished researcher in the field even say “I am not interested in stupid implementation”, as if she could just think up a structure and leave it to a few graduate students to implement.