Sent to you by Mike via Google Reader: Accounting for Numbers via Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics by Steve Landsburg on 11/13/12
Over at Less Wrong, the estimable Eliezer Yudkowsky attempts to account for the meaning of statements in arithmetic and the ontological status of numbers. I started to post a comment, but it got long enough that I’ve turned my comment into a blog post. I’ve tried to summarize my understanding of Yudkowsky’s position along the way, but of course it’s possible I’ve gotten something wrong.
It’s worth noting that every single point below is something I’ve blogged about before. At the moment I’m too lazy to insert links to all those earlier blog posts, but I might come back and put the links in later. In any event, I think this post stands alone. Because it got long, I’ve inserted section numbers for the convenience of commenters who might want to refer to particular passages.
1. Yudkowsky poses, in essence, the following question:
Main Question, My Version: In what sense is the sentence “two plus two equals four” meaningful and/or true?
Yudkowsky phrases the question a little differently. What he actually asks is:
Main Question, Original Version: In what sense is the sentence “2 + 2 = 4″ meaningful and/or true?”
This, I think, threatens to confuse the issue. It’s important to distinguish between the numeral “2″, which is a formal symbol designed to be manipulated according to formal rules, and the noun “two”, which appears to name something, namely a particular number. Because Yudkowsky is asking about meaning and truth, I presume it is the noun, and not the symbol, that he intends to mention. So I’ll stick with my version, and translate his remarks accordingly.
2. Yudkowsky provisionally offers the following answer:
First Provisional Answer: The sentence “two plus two equals four” means that the expression “2 + 2 = 4″ is a valid inference from the axioms of Peano arithmetic.
He then provisionally rejects this provisional answer on the grounds (with which I wholeheartedly agree) that “figuring out facts about the natural numbers doesn’t feel like the operation of making up assumptions and then deducing conclusions from them.” He goes on to say: “It feels like the numbers are just out there, and the only point of making up the axioms of Peano Arithmetic was to allow mathematicians to talk about them.”
He’s certainly right that it feels — to me, and, I am sure to almost everyone who has ever thought much about arithmetic — like the numbers are just “out there”. On the other hand, I’d quibble with Yudkowsky’s assessment of the point of Peano arithmetic. The point isn’t to “allow mathematicians to talk” about numbers; mathematicians from Pythagoras through Dedekind had absolutely no problem talking about numbers in the absence of the Peano axioms. Instead, the point of the Peano axioms was to model what mathematicians do when they’re talking about numbers. Like all good models, the Peano axioms are a simplification that captures important aspects of reality without attempting to reproduce reality in detail.
3. To reach a closer understanding of what numbers are, Yudkowsky imagines trying to explain them to a logician with a full grasp of logic but no grasp of numbers. Here I think Yudkowsky has fooled himself into imagining an impossibility. If you grasp logic, you grasp the idea of a proof. If you grasp the idea of a proof, you grasp the idea of a sequence of logical steps. If you grasp the idea of a sequence of logical steps, you grasp the idea of a sequence. If you grasp the idea of a sequence, you already know a lot about numbers. This is one reason why I believe that any attempt to account for numbers via logic must ultimately be circular.
4. Be that as it may, Yudkowsky goes on to try to explain to his fictional interlocutor what numbers are. He begins by essentially stating the first order Peano axioms: 0 is a number, every number has a successor, no two numbers have the same successor, and so forth. Eventually, he realizes that this approach isn’t taking him quite where he wants to go and makes a bit of a course correction (as we’ll see below). But I think more than a course correction is called for; he’s gone off in entirely the wrong direction. He’s listing the properties of numbers, but not even trying to explain what they are. If I were explaining numbers to a naif, I’d probably start with something like Bertrand Russell’s account of numbers: We say that two sets of objects are “equinumerous” if they can be placed in one-one correspondence with each other; a “number” is that which all sets equinumerous to a given set have in common. Whether or not that works in detail, it’s at least an attempt at a definition, as opposed to a mere list of properties.
5. Yudkowsky, in his fictional conversation with his fictional logician, eventually comes to realize that neither the first order Peano axioms nor any other first-order system can uniquely characterize the natural numbers. This is a consequence of Godel’s Incompleteness Theorem, or even more fundamentally of the Lowenheim-Skolem Theorem. What it means is that no matter what axioms you start with, there are going to be multiple systems that satisfy those axioms; the natural numbers are only one of those systems, so your axioms cannot collectively specify the natural numbers.
6. Yudkowsky solves his problem by passing to second order Peano arithmetic — “second order” meaning that, in addition to using variables to represent numbers, you can also use variables to to represent sets of numbers. He correctly notes that second order Peano arithmetic has a unique model. (I am using the word “model” here in the technical sense of logic, not in the informal social-sciencey sort of way that I used it in point 2 above.) This means that sure enough, there is one and only one system that satisfies all the axioms of second-order arithmetic. And he concludes that:
Y’s conclusion: That’s why the mathematical study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms.
But this is disastrously wrong for at least two reasons, each of which deserves its own numbered point.
7. Yudkowsky leaps from “the natural numbers can be precisely specified by second order logic” to “the .. study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms”. This is wrong, wrong, wrong, because second order logic is not logic. Indeed, the whole point of logic is that it is a mechanical system for deriving inferences from assumptions, based on the forms of sentences without any reference to their meanings. (Thus if we assume that all bachelors are unmarried and that Walter is a bachelor, we can infer that Walter is unmarried, without having to know anything at all about who walter is, or what the words “bachelor” and ‘unmarried” mean.) That’s why you’re not allowed to set up an axiom system in which all the true theorems of arithmetic are taken as axioms — there is no mechanical procedure for determining whether a given statement is or is not a true theorem of arithmetic (see Tarski’s theorem on the undefinability of truth) and therefore no mechanical procedure for determining what is or is not an axiom in that system. In second-order Peano arithemetic, we have an analogous problem: The axioms can be identified mechanically, but the rules of inference can’t. A properly programmed computer can examine a first-order proof and tell you if it’s valid or not; that is, it can tell you whether each step does in fact follow logically from some of the previous steps. But no computer can do the same for second-order proofs.
So the study of second-order consequences is not logic at all; to tease out all the second-order consequences of your second-order axioms, you need to confront not just the forms of sentences but their meanings. In other words, you have to understand meanings before you can carry out the operation of inference. But Yudkowsky is trying to derive meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.
8. Even putting all that aside, Yudkowsky is relying on a theorem when he says that second-order Peano arithmetic has a unique model. That theorem requires a substantial dose of set theory. So in order to avoid taking numbers as primitive objects, he’s effectively resorted to taking sets as primitive objects. But why is it any more satisfying to take set theory as “given” than to take numbers as “given”? Indeed, the formal study of numbers precedes the formal study of sets by millennia, which suggests that numbers are a more natural starting point than sets are. Whether or not you buy that argument, it’s important to recognize that Yudkowsky has “solved” the problem of accounting for numbers only by reducing it to the problem of accounting for sets — except that he hasn’t even done that, because his reduction relies on pretending that second order logic is logic.
9. All of which leaves us with the problem of accounting for numbers, and for the meaning of statements like “two plus two equals four”. To me, by far the most satisfying solution is a full-fledged Platonic acknowledgement that numbers are indeed just “out there” and that they are directly accessible to our intuitions. I embrace this view for (at least) three reasons: A. After a lifetime of thinking about numbers, it feels right to me. B. Pretty much every one else who spends his/her life thinking about numbers has come to the same conclusion. C. It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”, partly because there is in fact a unique system of (standard) natural numbers, whereas the properties of the physical universe appear to be far more contingent and therefore unnecessary. I’ve given an account in The Big Questions of how the existence of numbers can account for the existence of the physical universe; I think it would be very difficult to go in the opposite direction (though I’ve seen some pretty good attempts). Therefore, accepting numbers as primary and accounting for the universe as a necessary consequence of numbers seems to me to be the ontologically parsimonious thing to do, and I like parsimony.
10. Needless to say, point 9 is not a proof. But I know of no alternative story that strikes me as even remotely plausible. Moreover, the alternative stories all seem to go wrong in pretty much the same ways; for example, every single point above is one I’ve blogged about before in other contexts, but here they are, all being relevant again.
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Weirdly, Landsburg seems to totally overlook the fact that most of us have been trained, for decades, to think of numbers in a particular way. It may be parsimonious to think of numbers as existing "out there", but it's probably even more parsimonious to think of them the way we think of hopscotch, namely as some mechanical process common between people, possessing fad-like qualities, that we may not remember learning.
Surely, Lansburg would smugly object that people have thought about numbers for longer than there have been public school systems, and this is true enough. But David Graeber has pointed out (see his book on debt) that not only mathematics, but philosophy as well, apparently emerged (in Rome, India, and China, around the same time) *in response* to coinage. It seems that nobody particularly cared that 2+2=4 until there were men with sharp things willing to stab you if you didn't give them exactly 4 somethings. In other words, the money system itself is an indoctrination into a particular way of thinking about numbers and logic.
See, for example, the Pirahã people, who, contrary to Lansburg's assertion that the natural numbers are "directly accessible", have no number system at all, and appear to have lost the ability to acquire one by adulthood. They have sought outside help in understanding mathematics only because of concerns about being ripped off by their neighbors in trade.
Then again, I've always been a bit skeptical of the debate about the ontology of mathematics, which strikes me as rather like analyzing the political opinions of a corpse. The subtext here is: We already agree about the details of mathematics, we just need to discuss what tie it should wear when we take it to meet Mother. But there are far deeper potential disagreements within mathematics itself, many of which would spring to life if we loosened the stranglehold that the hegemonic university system holds on career and research.
On Wed, Nov 14, 2012 at 6:02 AM, Mike michaelbish@gmail.com wrote:
Sent to you by Mike via Google Reader:
Accounting for Numbershttp://www.thebigquestions.com/2012/11/14/accounting-for-numbers/ via Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physicshttp://www.thebigquestions.comby Steve Landsburg on 11/13/12
Over at Less Wrong http://lesswrong.com/lw/f4e/logical_pinpointing/, the estimable Eliezer Yudkowskyhttp://en.wikipedia.org/wiki/Eliezer_yudkowskyattempts to account for the meaning of statements in arithmetic and the ontological status of numbers. I started to post a comment, but it got long enough that I’ve turned my comment into a blog post. I’ve tried to summarize my understanding of Yudkowsky’s position along the way, but of course it’s possible I’ve gotten something wrong.
It’s worth noting that every single point below is something I’ve blogged about before. At the moment I’m too lazy to insert links to all those earlier blog posts, but I might come back and put the links in later. In any event, I think this post stands alone. Because it got long, I’ve inserted section numbers for the convenience of commenters who might want to refer to particular passages.
- Yudkowsky poses, in essence, the following question:
*Main Question, My Version:* *In what sense is the sentence “two plus two equals four” meaningful and/or true?*
Yudkowsky phrases the question a little differently. What he actually asks is:
*Main Question, Original Version:* *In what sense is the sentence “2 + 2 = 4″ meaningful and/or true?”*
This, I think, threatens to confuse the issue. It’s important to distinguish between the numeral “2″, which is a formal symbol designed to be manipulated according to formal rules, and the noun “two”, which appears to name something, namely a particular number. Because Yudkowsky is asking about meaning and truth, I presume it is the noun, and not the symbol, that he intends to mention. So I’ll stick with my version, and translate his remarks accordingly.
- Yudkowsky provisionally offers the following answer:
*First Provisional Answer:* *The sentence “two plus two equals four” means that the expression “2 + 2 = 4″ is a valid inference from the axioms of Peano arithmetic.*
He then provisionally rejects this provisional answer on the grounds (with which I wholeheartedly agree) that “figuring out facts about the natural numbers doesn’t feel like the operation of making up assumptions and then deducing conclusions from them.” He goes on to say: “It feels like the numbers are just *out* there, and the only point of making up the axioms of Peano Arithmetic was to allow mathematicians to talk about them.”
He’s certainly right that it feels — to me, and, I am sure to almost everyone who has ever thought much about arithmetic — like the numbers are just “out there”. On the other hand, I’d quibble with Yudkowsky’s assessment of the point of Peano arithmetic. The point isn’t to “allow mathematicians to talk” about numbers; mathematicians from Pythagoras through Dedekind had absolutely no problem talking about numbers in the absence of the Peano axioms. Instead, the point of the Peano axioms was to *model* what mathematicians do when they’re talking about numbers. Like all good models, the Peano axioms are a simplification that captures important aspects of reality without attempting to reproduce reality in detail.
- To reach a closer understanding of what numbers *are*, Yudkowsky
imagines trying to explain them to a logician with a full grasp of logic but no grasp of numbers. Here I think Yudkowsky has fooled himself into imagining an impossibility. If you grasp logic, you grasp the idea of a proof. If you grasp the idea of a proof, you grasp the idea of a sequence of logical steps. If you grasp the idea of a sequence of logical steps, you grasp the idea of a sequence. If you grasp the idea of a sequence, you already know a lot about numbers. This is one reason why I believe that any attempt to account for numbers via logic must ultimately be circular.
- Be that as it may, Yudkowsky goes on to try to explain to his fictional
interlocutor what numbers are. He begins by essentially stating the first order Peano axioms: 0 is a number, every number has a successor, no two numbers have the same successor, and so forth. Eventually, he realizes that this approach isn’t taking him quite where he wants to go and makes a bit of a course correction (as we’ll see below). But I think more than a course correction is called for; he’s gone off in entirely the wrong direction. He’s listing the *properties* of numbers, but not even *trying* to explain what they *are*. If I were explaining numbers to a naif, I’d probably start with something like Bertrand Russell’s account of numbers: We say that two sets of objects are “equinumerous” if they can be placed in one-one correspondence with each other; a “number” is that which all sets equinumerous to a given set have in common. Whether or not that works in detail, it’s at least an attempt at a definition, as opposed to a mere list of properties.
- Yudkowsky, in his fictional conversation with his fictional logician,
eventually comes to realize that neither the first order Peano axioms nor any other first-order system can uniquely characterize the natural numbers. This is a consequence of Godel’s Incompleteness Theoremhttp://www.thebigquestions.com/2009/11/25/godel-in-a-nutshell/, or even more fundamentally of the Lowenheim-Skolem Theorem. What it means is that no matter what axioms you start with, there are going to be multiple systems that satisfy those axioms; the natural numbers are only one of those systems, so your axioms cannot collectively specify the natural numbers.
- Yudkowsky solves his problem by passing to second order Peano
arithmetic — “second order” meaning that, in addition to using variables to represent numbers, you can also use variables to to represent *sets* of numbers. He correctly notes that second order Peano arithmetic has a unique model. (I am using the word “model” here in the technical sense of logic, not in the informal social-sciencey sort of way that I used it in point 2 above.) This means that sure enough, there is one and only one system that satisfies all the axioms of second-order arithmetic. And he concludes that:
*Y’s conclusion*: *That’s why the mathematical study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms.*
But this is disastrously wrong for at least two reasons, each of which deserves its own numbered point.
- Yudkowsky leaps from “the natural numbers can be precisely specified by
second order logic” to “the .. study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms”. This is wrong, wrong, wrong, because *second order logic is not logic*http://www.thebigquestions.com/2010/07/27/first-things-and-second-things/. Indeed, the whole point of logic is that it is a *mechanical* system for deriving inferences from assumptions, based on the *forms* of sentences without any reference to their *meanings*. (Thus if we assume that all bachelors are unmarried and that Walter is a bachelor, we can infer that Walter is unmarried, without having to know anything at all about who walter is, or what the words “bachelor” and ‘unmarried” mean.) That’s why you’re not allowed to set up an axiom system in which all the true theorems of arithmetic are taken as axioms — there is no mechanical procedure for determining whether a given statement is or is not a true theorem of arithmetic (see Tarski’s theorem on the undefinability of truth) and therefore no mechanical procedure for determining what is or is not an axiom in that system. In second-order Peano arithemetic, we have an analogous problem: The *axioms* can be identified mechanically, but the *rules of inference* can’t. A properly programmed computer can examine a first-order proof and tell you if it’s valid or not; that is, it can tell you whether each step does in fact follow logically from some of the previous steps. But no computer can do the same for second-order proofs.
So the study of second-order consequences is not logic at all; to tease out all the second-order consequences of your second-order axioms, you need to confront not just the *forms* of sentences but their *meanings*. In other words, *you have to understand meanings before you can carry out the operation of inference*. But Yudkowsky is trying to *derive* meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.
- Even putting all that aside, Yudkowsky is relying on a *theorem* when
he says that second-order Peano arithmetic has a unique model. That theorem requires a substantial dose of set theory. So in order to avoid taking numbers as primitive objects, he’s effectively resorted to taking *sets*as primitive objects. But why is it any more satisfying to take set theory as “given” than to take numbers as “given”? Indeed, the formal study of numbers precedes the formal study of sets by millennia, which suggests that numbers are a more natural starting point than sets are. Whether or not you buy that argument, it’s important to recognize that Yudkowsky has “solved” the problem of accounting for numbers only by reducing it to the problem of accounting for sets — except that he hasn’t even done that, because his reduction relies on pretending that second order logic is logic.
- All of which leaves us with the problem of accounting for numbers, and
for the meaning of statements like “two plus two equals four”. To me, by far the most satisfying solution is a full-fledged Platonic acknowledgement http://www.thebigquestions.com/2010/01/13/real-numbers/that numbers are indeed just “out there” and that they are directly accessible to our intuitions. I embrace this view for (at least) three reasons: A. After a lifetime of thinking about numbers, it feels right to me. B. Pretty much every one else who spends his/her life thinking about numbers has come to the same conclusion. C. It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”, partly because there is in fact a unique system of (standard) natural numbers, whereas the properties of the physical universe appear to be far more contingent and therefore unnecessary. I’ve given an account in *The Big Questions*http://www.amazon.com/Big-Questions-Philosophy-Mathematics-Economics/dp/143914821X/ref=nosim/?tag=moseissase-20of how the existence of numbers can account for the existence of the physical universe; I think it would be very difficult to go in the opposite direction (though I’ve seen some pretty good attempts). Therefore, accepting numbers as primaryhttp://www.thebigquestions.com/2010/12/02/first-things/and accounting for the universe as a necessary consequence of numbers seems to me to be the ontologically parsimonioushttp://www.thebigquestions.com/2012/09/18/wwct-what-would-copernicus-think/thing to do, and I like parsimony.
- Needless to say, point 9 is not a proof. But I know of no alternative
story that strikes me as even remotely plausible. Moreover, the alternative stories all seem to go wrong in pretty much the same ways; for example, every single point above is one I’ve bloggedhttp://www.thebigquestions.com/2010/09/01/basic-arithmetic-on-what-there-is/ abouthttp://www.thebigquestions.com/2010/08/18/basic-arithmetic-part-ii/ beforehttp://www.thebigquestions.com/2010/08/19/basic-arithmetic-part-iii-the-map-is-not-the-territory/in other contexts, but here they are, all being relevant again.
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On Wed, Nov 14, 2012 at 11:47 AM, Andrew Dudzik adudzik@gmail.com wrote:
Weirdly, Landsburg seems to totally overlook the fact that most of us have been trained, for decades, to think of numbers in a particular way. It may be parsimonious to think of numbers as existing "out there", but it's probably even more parsimonious to think of them the way we think of hopscotch, namely as some mechanical process common between people, possessing fad-like qualities, that we may not remember learning.
Surely, Lansburg would smugly object that people have thought about numbers for longer than there have been public school systems, and this is true enough. But David Graeber has pointed out (see his book on debt) that not only mathematics, but philosophy as well, apparently emerged (in Rome, India, and China, around the same time) *in response* to coinage. It seems that nobody particularly cared that 2+2=4 until there were men with sharp things willing to stab you if you didn't give them exactly 4 somethings. In other words, the money system itself is an indoctrination into a particular way of thinking about numbers and logic.
See, for example, the Pirahã people, who, contrary to Lansburg's assertion that the natural numbers are "directly accessible", have no number system at all, and appear to have lost the ability to acquire one by adulthood. They have sought outside help in understanding mathematics only because of concerns about being ripped off by their neighbors in trade.
There is some neat research on how our innate number systems are more logarithmic than linear by intuition (I believe this gets discussed in *Thinking Fast and Slow*). We're good at estimating orders of magnitude but need to learn how to add beyond very small numbers. You can see this when you ask people to compare numbers that are outside their experience, as when politicians exploit the confusability of billions and millions or when you try to talk about evolutionary time scales with most people.
Then again, I've always been a bit skeptical of the debate about the
ontology of mathematics, which strikes me as rather like analyzing the political opinions of a corpse. The subtext here is: We already agree about the details of mathematics, we just need to discuss what tie it should wear when we take it to meet Mother. But there are far deeper potential disagreements within mathematics itself, many of which would spring to life if we loosened the stranglehold that the hegemonic university system holds on career and research.
I'd love to hear some more thoughts on this.
On Wed, Nov 14, 2012 at 6:02 AM, Mike michaelbish@gmail.com wrote:
Sent to you by Mike via Google Reader:
Accounting for Numbershttp://www.thebigquestions.com/2012/11/14/accounting-for-numbers/ via Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physicshttp://www.thebigquestions.comby Steve Landsburg on 11/13/12
Over at Less Wrong http://lesswrong.com/lw/f4e/logical_pinpointing/, the estimable Eliezer Yudkowskyhttp://en.wikipedia.org/wiki/Eliezer_yudkowskyattempts to account for the meaning of statements in arithmetic and the ontological status of numbers. I started to post a comment, but it got long enough that I’ve turned my comment into a blog post. I’ve tried to summarize my understanding of Yudkowsky’s position along the way, but of course it’s possible I’ve gotten something wrong.
It’s worth noting that every single point below is something I’ve blogged about before. At the moment I’m too lazy to insert links to all those earlier blog posts, but I might come back and put the links in later. In any event, I think this post stands alone. Because it got long, I’ve inserted section numbers for the convenience of commenters who might want to refer to particular passages.
- Yudkowsky poses, in essence, the following question:
*Main Question, My Version:* *In what sense is the sentence “two plus two equals four” meaningful and/or true?*
Yudkowsky phrases the question a little differently. What he actually asks is:
*Main Question, Original Version:* *In what sense is the sentence “2 + 2 = 4″ meaningful and/or true?”*
This, I think, threatens to confuse the issue. It’s important to distinguish between the numeral “2″, which is a formal symbol designed to be manipulated according to formal rules, and the noun “two”, which appears to name something, namely a particular number. Because Yudkowsky is asking about meaning and truth, I presume it is the noun, and not the symbol, that he intends to mention. So I’ll stick with my version, and translate his remarks accordingly.
- Yudkowsky provisionally offers the following answer:
*First Provisional Answer:* *The sentence “two plus two equals four” means that the expression “2 + 2 = 4″ is a valid inference from the axioms of Peano arithmetic.*
He then provisionally rejects this provisional answer on the grounds (with which I wholeheartedly agree) that “figuring out facts about the natural numbers doesn’t feel like the operation of making up assumptions and then deducing conclusions from them.” He goes on to say: “It feels like the numbers are just *out* there, and the only point of making up the axioms of Peano Arithmetic was to allow mathematicians to talk about them.”
He’s certainly right that it feels — to me, and, I am sure to almost everyone who has ever thought much about arithmetic — like the numbers are just “out there”. On the other hand, I’d quibble with Yudkowsky’s assessment of the point of Peano arithmetic. The point isn’t to “allow mathematicians to talk” about numbers; mathematicians from Pythagoras through Dedekind had absolutely no problem talking about numbers in the absence of the Peano axioms. Instead, the point of the Peano axioms was to *model* what mathematicians do when they’re talking about numbers. Like all good models, the Peano axioms are a simplification that captures important aspects of reality without attempting to reproduce reality in detail.
- To reach a closer understanding of what numbers *are*, Yudkowsky
imagines trying to explain them to a logician with a full grasp of logic but no grasp of numbers. Here I think Yudkowsky has fooled himself into imagining an impossibility. If you grasp logic, you grasp the idea of a proof. If you grasp the idea of a proof, you grasp the idea of a sequence of logical steps. If you grasp the idea of a sequence of logical steps, you grasp the idea of a sequence. If you grasp the idea of a sequence, you already know a lot about numbers. This is one reason why I believe that any attempt to account for numbers via logic must ultimately be circular.
- Be that as it may, Yudkowsky goes on to try to explain to his
fictional interlocutor what numbers are. He begins by essentially stating the first order Peano axioms: 0 is a number, every number has a successor, no two numbers have the same successor, and so forth. Eventually, he realizes that this approach isn’t taking him quite where he wants to go and makes a bit of a course correction (as we’ll see below). But I think more than a course correction is called for; he’s gone off in entirely the wrong direction. He’s listing the *properties* of numbers, but not even *trying
- to explain what they *are*. If I were explaining numbers to a naif,
I’d probably start with something like Bertrand Russell’s account of numbers: We say that two sets of objects are “equinumerous” if they can be placed in one-one correspondence with each other; a “number” is that which all sets equinumerous to a given set have in common. Whether or not that works in detail, it’s at least an attempt at a definition, as opposed to a mere list of properties.
- Yudkowsky, in his fictional conversation with his fictional logician,
eventually comes to realize that neither the first order Peano axioms nor any other first-order system can uniquely characterize the natural numbers. This is a consequence of Godel’s Incompleteness Theoremhttp://www.thebigquestions.com/2009/11/25/godel-in-a-nutshell/, or even more fundamentally of the Lowenheim-Skolem Theorem. What it means is that no matter what axioms you start with, there are going to be multiple systems that satisfy those axioms; the natural numbers are only one of those systems, so your axioms cannot collectively specify the natural numbers.
- Yudkowsky solves his problem by passing to second order Peano
arithmetic — “second order” meaning that, in addition to using variables to represent numbers, you can also use variables to to represent *sets* of numbers. He correctly notes that second order Peano arithmetic has a unique model. (I am using the word “model” here in the technical sense of logic, not in the informal social-sciencey sort of way that I used it in point 2 above.) This means that sure enough, there is one and only one system that satisfies all the axioms of second-order arithmetic. And he concludes that:
*Y’s conclusion*: *That’s why the mathematical study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms.*
But this is disastrously wrong for at least two reasons, each of which deserves its own numbered point.
- Yudkowsky leaps from “the natural numbers can be precisely specified
by second order logic” to “the .. study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms”. This is wrong, wrong, wrong, because *second order logic is not logic*http://www.thebigquestions.com/2010/07/27/first-things-and-second-things/. Indeed, the whole point of logic is that it is a *mechanical* system for deriving inferences from assumptions, based on the *forms* of sentences without any reference to their *meanings*. (Thus if we assume that all bachelors are unmarried and that Walter is a bachelor, we can infer that Walter is unmarried, without having to know anything at all about who walter is, or what the words “bachelor” and ‘unmarried” mean.) That’s why you’re not allowed to set up an axiom system in which all the true theorems of arithmetic are taken as axioms — there is no mechanical procedure for determining whether a given statement is or is not a true theorem of arithmetic (see Tarski’s theorem on the undefinability of truth) and therefore no mechanical procedure for determining what is or is not an axiom in that system. In second-order Peano arithemetic, we have an analogous problem: The *axioms* can be identified mechanically, but the *rules of inference* can’t. A properly programmed computer can examine a first-order proof and tell you if it’s valid or not; that is, it can tell you whether each step does in fact follow logically from some of the previous steps. But no computer can do the same for second-order proofs.
So the study of second-order consequences is not logic at all; to tease out all the second-order consequences of your second-order axioms, you need to confront not just the *forms* of sentences but their *meanings*. In other words, *you have to understand meanings before you can carry out the operation of inference*. But Yudkowsky is trying to *derive* meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.
- Even putting all that aside, Yudkowsky is relying on a *theorem* when
he says that second-order Peano arithmetic has a unique model. That theorem requires a substantial dose of set theory. So in order to avoid taking numbers as primitive objects, he’s effectively resorted to taking *sets*as primitive objects. But why is it any more satisfying to take set theory as “given” than to take numbers as “given”? Indeed, the formal study of numbers precedes the formal study of sets by millennia, which suggests that numbers are a more natural starting point than sets are. Whether or not you buy that argument, it’s important to recognize that Yudkowsky has “solved” the problem of accounting for numbers only by reducing it to the problem of accounting for sets — except that he hasn’t even done that, because his reduction relies on pretending that second order logic is logic.
- All of which leaves us with the problem of accounting for numbers, and
for the meaning of statements like “two plus two equals four”. To me, by far the most satisfying solution is a full-fledged Platonic acknowledgement http://www.thebigquestions.com/2010/01/13/real-numbers/that numbers are indeed just “out there” and that they are directly accessible to our intuitions. I embrace this view for (at least) three reasons: A. After a lifetime of thinking about numbers, it feels right to me. B. Pretty much every one else who spends his/her life thinking about numbers has come to the same conclusion. C. It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”, partly because there is in fact a unique system of (standard) natural numbers, whereas the properties of the physical universe appear to be far more contingent and therefore unnecessary. I’ve given an account in *The Big Questions*http://www.amazon.com/Big-Questions-Philosophy-Mathematics-Economics/dp/143914821X/ref=nosim/?tag=moseissase-20of how the existence of numbers can account for the existence of the physical universe; I think it would be very difficult to go in the opposite direction (though I’ve seen some pretty good attempts). Therefore, accepting numbers as primaryhttp://www.thebigquestions.com/2010/12/02/first-things/and accounting for the universe as a necessary consequence of numbers seems to me to be the ontologically parsimonioushttp://www.thebigquestions.com/2012/09/18/wwct-what-would-copernicus-think/thing to do, and I like parsimony.
- Needless to say, point 9 is not a proof. But I know of no alternative
story that strikes me as even remotely plausible. Moreover, the alternative stories all seem to go wrong in pretty much the same ways; for example, every single point above is one I’ve bloggedhttp://www.thebigquestions.com/2010/09/01/basic-arithmetic-on-what-there-is/ abouthttp://www.thebigquestions.com/2010/08/18/basic-arithmetic-part-ii/ beforehttp://www.thebigquestions.com/2010/08/19/basic-arithmetic-part-iii-the-map-is-not-the-territory/in other contexts, but here they are, all being relevant again.
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I'm actually about halfway through *Thinking Fast and Slow* at the moment.
I'd love to hear some more thoughts on this.
This is something I'm currently thinking about a lot, so you might want to ask me again in a few months. But sections 4 and 5 of this essayhttp://www.law.harvard.edu/faculty/unger/english/pdfs/reorientation6.pdfon pragmatism by Roberto Unger, my favorite thinker, contains some choice quotations. Unger bemoans "the overwhelming influence that the disciplinary structure of the modern university, with its obsessive focus on the filtering out of unreliable people and ideas, has had upon the way we do science." But, as with most of Unger's work, you'll have to find the details elsewhere.
That essay, incidentally, is where I got the idea that the realism vs. constructivism debate is a pretty lie, covering up deeper, more unsettling concerns about how mathematics is actually practiced. Unger calls it a way of "keeping disagreement within the family", and notes that it is by no means confined to mathematics.
Related to the project of restructuring the social organization of the university system is the project of reorganizing the language surrounding science. Bruno Latour's excellent book *Politics of Naturehttp://en.wikipedia.org/wiki/Politics_of_Nature * attempts to do just that, though I've just barely begun reading it.
Thanks for your thoughts Andrew. I know enough to know that I can't evaluate the ideas Landsburg presents. I go back and forth on the value of philosophy more generally even the parts of it which I find appealing.
As I've expressed on readings before, I am highly skeptical of Latour's value. To cite an example of what bugs me from your link to the Wikipedia page on Politics of Nature: "Latour argues that this distinction between facts and values is rarely useful and in many situations dangerous. He claims that it leads to a system that ignores nature's socially constructed status and creates a political order without "due process of individual will"."
I'm aware that the fact-value distinction has some problems in a deep philosophical sense. But we just finished a presidential election in which, as usual, IMO, lying about important things proves to be a good political strategy. It wouldn't bother me so much if the lies were about things that are hard to prove - I expect that. But even fairly obvious lies, e.g. Romney's budget numbers, make good politics. This really damages my confidence in democracy. So when I hear that Latour's biggest concern about our political life is that scientists with their emphasis on facts are undermining democracy I am unimpressed by his insight.
On Wed, Nov 14, 2012 at 11:54 AM, Andrew Dudzik adudzik@gmail.com wrote:
I'm actually about halfway through *Thinking Fast and Slow* at the moment.
I'd love to hear some more thoughts on this.
This is something I'm currently thinking about a lot, so you might want to ask me again in a few months. But sections 4 and 5 of this essayhttp://www.law.harvard.edu/faculty/unger/english/pdfs/reorientation6.pdfon pragmatism by Roberto Unger, my favorite thinker, contains some choice quotations. Unger bemoans "the overwhelming influence that the disciplinary structure of the modern university, with its obsessive focus on the filtering out of unreliable people and ideas, has had upon the way we do science." But, as with most of Unger's work, you'll have to find the details elsewhere.
That essay, incidentally, is where I got the idea that the realism vs. constructivism debate is a pretty lie, covering up deeper, more unsettling concerns about how mathematics is actually practiced. Unger calls it a way of "keeping disagreement within the family", and notes that it is by no means confined to mathematics.
Related to the project of restructuring the social organization of the university system is the project of reorganizing the language surrounding science. Bruno Latour's excellent book *Politics of Naturehttp://en.wikipedia.org/wiki/Politics_of_Nature
- attempts to do just that, though I've just barely begun reading it.
Math mailing list Math@moomers.org http://mailman.moomers.org/mailman/listinfo/math
The ironic thing is that Latour is attempting to address precisely the problem you are complaining about (see the preamble to his essay on criticism http://www.pathguy.com/Latour.pdf). But one of the disturbing strengths of Rationalism is its insistence that everything that does not resemble itself must be in some sense a perversion of reality. Yet it has failed to adequately take into account its own historical biases, rooted in 17th century post-colonialism, and based more on a concept of power than on a concept of truth.
In any case, a system that presupposes a singular truth administered from on high is every bit as destructive as a system that leaves truth entirely up for grabs. In fact, the two have a tendency to transform into each other.
Rationalists who blame democracy are failing to first remove the plank from their own eye. The world has good reason to reject the technocracy that science is frequently caught wishing that it was. The question is not how to force the world to accept our particular version of natural reality; it is how best to assist others in pressing forward, towards greater solidarity and appreciation of the essential matters of concern. Yet somehow anyone who actually attempts such a reconstruction is ridiculed by the defenders of science, a sign that science itself has fallen into a kind of despair.
...for those getting tired of this philosophizing, I'll try to send some actual math out to the list later today to make up for it...
On Wed, Nov 14, 2012 at 11:19 AM, Michael Bishop michaelbish@gmail.comwrote:
Thanks for your thoughts Andrew. I know enough to know that I can't evaluate the ideas Landsburg presents. I go back and forth on the value of philosophy more generally even the parts of it which I find appealing.
As I've expressed on readings before, I am highly skeptical of Latour's value. To cite an example of what bugs me from your link to the Wikipedia page on Politics of Nature: "Latour argues that this distinction between facts and values is rarely useful and in many situations dangerous. He claims that it leads to a system that ignores nature's socially constructed status and creates a political order without "due process of individual will"."
I'm aware that the fact-value distinction has some problems in a deep philosophical sense. But we just finished a presidential election in which, as usual, IMO, lying about important things proves to be a good political strategy. It wouldn't bother me so much if the lies were about things that are hard to prove - I expect that. But even fairly obvious lies, e.g. Romney's budget numbers, make good politics. This really damages my confidence in democracy. So when I hear that Latour's biggest concern about our political life is that scientists with their emphasis on facts are undermining democracy I am unimpressed by his insight.
On Wed, Nov 14, 2012 at 11:54 AM, Andrew Dudzik adudzik@gmail.com wrote:
I'm actually about halfway through *Thinking Fast and Slow* at the moment.
I'd love to hear some more thoughts on this.
This is something I'm currently thinking about a lot, so you might want to ask me again in a few months. But sections 4 and 5 of this essayhttp://www.law.harvard.edu/faculty/unger/english/pdfs/reorientation6.pdfon pragmatism by Roberto Unger, my favorite thinker, contains some choice quotations. Unger bemoans "the overwhelming influence that the disciplinary structure of the modern university, with its obsessive focus on the filtering out of unreliable people and ideas, has had upon the way we do science." But, as with most of Unger's work, you'll have to find the details elsewhere.
That essay, incidentally, is where I got the idea that the realism vs. constructivism debate is a pretty lie, covering up deeper, more unsettling concerns about how mathematics is actually practiced. Unger calls it a way of "keeping disagreement within the family", and notes that it is by no means confined to mathematics.
Related to the project of restructuring the social organization of the university system is the project of reorganizing the language surrounding science. Bruno Latour's excellent book *Politics of Naturehttp://en.wikipedia.org/wiki/Politics_of_Nature
- attempts to do just that, though I've just barely begun reading it.
Math mailing list Math@moomers.org http://mailman.moomers.org/mailman/listinfo/math
participants (4)
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Andrew Dudzik -
Max Shron -
Michael Bishop -
Mike