Voted up because this is a great topic that I’d like us to try and begin to tackle.
But this post really frustrating to try to respond to. Not because it is especially wrong-headed or poorly written but just because it is a little hard for me to find my way around your theory. It is difficult to find a point of traction. In general, I suspect it just isn’t really solving problems but eliding distinctions and ignoring problems (just based on what I do know and the relative shortness of this compared to most other work in philosophy of math). This is pretty much the way I feel about what Eliezer has said on the subject and just about every single thought I’ve ever had on the subject. I’m also not sure I’m familiar enough with the subject area to be able to examine this post in the way it requires.
So I suspect I’ll end up prodding you in a couple places but to begin with: what exactly do you take the Platonist thesis to be? If there is an analogical relationship between a particular expression in our system of inscriptions and our rules for manipulating them (i.e. a written equation) and a physical system (i.e. a system that equation describes) that seems to suggest an underlying structure which is instantiated in both the mathematical expression and the physical system. That such structures exist independently of the mind strikes me as a platonist position. What exactly is wrong with that position? Or what even did you say to contradict it?
Perhaps we need to have a discussion about abstract objects in general before tackling the math.
I do think you’re right about the map-territory confusions here. They definitely abound.
In general, I suspect it just isn’t really solving problems but eliding distinctions and ignoring problems (just based on what I do know and the relative shortness of this compared to most other work in philosophy of math).
This is not a good heuristic, because in philosophy, works tend to be longest when they’re confused, because most of the length tends to be spent repairing the damage caused by a mistake early on.
So philosophy can get long because the author is running damage control. True. But it can also be short because the author is trying to answer 5-6 questions at once without engaging with the arguments of those who argue against his position. So length by itself- maybe a bad heuristic. But I’m leveraging this heuristic with enough background to make it work.
what exactly do you take the Platonist thesis to be?
That there is an immaterial realm of ideal forms (structures, concepts) of which our universe consists solely of imperfect approximations of.
If there is an analogical relationship between a particular expression in our system of inscriptions and our rules for manipulating them (i.e. a written equation) and a physical system (i.e. a system that equation describes) that seems to suggest an underlying structure which is instantiated in both the mathematical expression and the physical system.That such structures exist independently of the mind strikes me as a platonist position.
I would say instead that there is some generating function for reality. A system of inscriptions/rules can describe that generating function imperfectly; but this in no way means that the rule/inscription system has some existence apart from its instantiation as the universe itself, and again explicitly in a model.
That there is an immaterial realm of ideal forms (structures, concepts) of which our universe consists solely of imperfect approximations of.
This stuff about imperfect approximations is just a remnant of Plato’s mysticism. Few modern platonists would say anything like that. This notion of an immaterial “realm” has similar connotations. How about:
Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental.
Platonism is appealing because it adheres to our norm of accepting the existence of things we make true statements about. “Silas is cool” implies the existence of Silas. Similarly, “3 is prime” implies the existence of 3. The list of non-platonist options as far as I can recall consists of: mathematical objects are mental objects, mathematical objects are physical objects, statements about mathematical objects are false (like statements about Santa Claus), or statements about mathematical objects are actually paraphrases of sentences that don’t commit us to the existence of abstract objects.
It seems like you are trying something like the last. But for this strategy you really should give explicit paraphrases or, ideally, a method for paraphrasing all mathematical truths.
I would say instead that there is some generating function for reality. A system of inscriptions/rules can describe that generating function imperfectly; but this in no way means that the rule/inscription system has some existence apart from its instantiation as the universe itself, and again explicitly in a model.
But then what kind of thing is this function? It clearly isn’t merely a set of inscriptions and rules for manipulating them (the models). Nor is it merely the physical universe. We talk like it exists. If it doesn’t, why do we talk like this and what do claims about it really mean?
At least for geometrical forms, the abstractions may be intrinsic to the mind, even if they don’t exist outside it.
In The Man Who Mistook His Wife for a Hat, there’s a description of a man who lost the ability to visually recognize ordinary objects, though he could still see. The one description suggest that he just saw geometry.
In Crashing Through, which is about a man who lost his sight at age 3 and recovered it in middle age and which has a lot about recovered vision and the amount of processing it takes to make sense of what you see, there’s mention of some people who are very disappointed when they recover their sight—they’re constantly comparing the world to an idea of it which is perfectly clean and geometrical.
In The Man Who Mistook His Wife for a Hat, there’s a description of a man who lost the ability to visually recognize ordinary objects, though he could still see. The one description suggest that he just saw geometry.
I’m a little confused: did is visual field lose focus such that, instead of seeing the details on objects and their imperfections he actually just saw idealized geometric figures?
One problem with this as evidence of the possibility that geometric forms could exist only in the human mind is that it presumably only applies to a rather narrow class of geometric forms. It would be weird if the geometric forms we have innate access to had a different ontological status from forms that can’t be instantiated in the human mind: like a 1000-sided polygon or something in 4+ dimensions.
What I meant was that, if people have simple geometric forms built deep into their minds, then it would be tempting to conclude that math has an objective eternal existence because it feels that way.
In any case, I found the actual quote, and I’ve very uncertain that it suggests what I thought it did. It seems as though the man was at least as sensitive to simple topology as geometry, However, people don’t romanticize topology.
I had stopped at a florist on my way to his apartment and bought myself an extravagant red rose for my buttonhole. Now I removed this and handed it to him. He took it like a botanist or morphologist given a specimen, not like person given a flower.
“About six inches in length,’ he commented. ‘A convoluted red form with a linear green attachment.’
‘Yes,’ I said encouragingly, ‘and what do you think it is, Dr P.?’
‘Not easy to say.’ He seemed perplexed. ‘It lacks the simple symmetry of the Platonic solids, although it may have a higher symmetry of its own… I think this could be an inflorescence or flower.’
‘Could be?’ I queried.
‘Could be,’ he confirmed.
‘Smell it,’ I suggested, and he again looked somewhat puzzled, as if I had asked him to smell a higher symmetry. But he complied courteously, and took it to his nose. Now, suddenly, he came to life.
‘Beautiful!’ he exclaimed. ‘An early rose. What a heavenly smell!’ He started to hum ‘Die Rose, die Lillie…’ Reality, it seemed, might by conveyed by smell, not by sight.
I tried one final test. It was still a cold day, in early spring, and I had thrown my coat and gloves on the sofa.
‘What is this?’ I asked, holding up a glove.
‘May I examine it?’ he asked, and, taking it from me, he proceeded to examine it as he had examined the geometrical shapes.
‘A continuous surface,’ he announced at last, ‘infolded on itself. It appears to have’ – he hesitated – ‘five outpouchings, if this is the word.’
It’s a wonderful extract in any case. It is fascinating to see someone describing the world without anything more than the phenomenology of his surroundings. It is interesting that the concepts he had access to were mathematical and geometric- that these concepts involve a part of the brain separate from the part that involves more complex and obviously learned concepts like shoe, glove, and flower does seem important to keep in mind when evaluating the evidence on this issue. You’re right that this fact could lead to us positing a false ontological difference… though of course there are those who will say “gloveness” and “flowerness” are abstract objects as well. The fact that these concepts are processed in different parts of the brain could also be taken as evidence for the distinction in that different evolutionary processes generated these two kinds of concepts. I’m not sure how to interpret this. Good for keeping in mind though.
In The Man Who Mistook His Wife for a Hat, there’s a description of a man who lost the ability to visually recognize ordinary objects, though he could still see. The one description suggest that he just saw geometry.
Googling it looks like maybe he just had visual agnosia? Which doesn’t really entail what you’re saying. That would mean that he could see normally but just couldn’t recognize figures as objects with names and functions. Or are you saying the details of objects disappeared and all that was left were the basic geometric forms?
On problem with this as evidence of the possibility that geometric forms could exist only in the human mind is that it presumably only applies to a rather narrow class of geometric forms. It would be weird if the geometric forms we have innate access to had a different ontological status from forms that can’t be instantiated in the human mind: like a 1000-sided polygon or something in 4+ dimensions.
Platonism is appealing because it adheres to our norm of accepting the existence of things we make true statements about. “Silas is cool” implies the existence of Silas.
To bring the comparison closer to the mark: It would also imply the existence of ‘cool’.
Heh. What I had in mind was Quine’s criterion for ontological commitment under which it wouldn’t. So Silas is cool is something like, where cool is the predicate letter C: ∃x(Cx ∩ x=”Silas”). We’re committed to the existence of the bound variables (to exist is to be the value of a bound variable) but not of the properties, there doesn’t have to be anything like coolness (assuming that was what you were suggesting).
There is an older argument that claims all words must refer to things and thus a word like “cool” must refer to coolness. But I wasn’t intending to make that argument (though I didn’t say nearly enough in my previous comment to expect everyone to figure that out).
We’re committed to the existence of the bound variables (to exist is to be the value of a bound variable) but not of the properties, there doesn’t have to be anything like coolness (assuming that was what you were suggesting).
My reading of Silas’s essay (and in particular looking at his diagrams) gave me impression that his ‘2’ is closer to what you would describe as a ‘property’ than the category in which you put ‘Silas’.
I was just starting from the observation that in our mathematical discourse we treat numbers like objects, not properties. “The number between 2 and 4”, “there is a prime number greater than one million”, “5 is odd” etc. all treat numbers as objects.
I would call those properties that had properties. But I’m a programmer, not a mathematician or philosopher (so don’t know which limitations I’m supposed to have placed around my thinking!)
By the way, I think ‘cool’ is kinda ‘lame’ but ‘awesomeness’ is kinda ‘cool’. Just sayin’.
Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental.
Platonism is appealing because it adheres to our norm of accepting the existence of things we make true statements about. “Silas is cool” implies the existence of Silas. Similarly, “3 is prime” implies the existence of 3.
This, I claim, is where you should stop the chain. You’ve erred from the moment you think that that a true statement implies independent existence of its predicates, leaving out the possibility that the predicates exist only as part of models. (Or, equivalently, from treating the term “existence” as having the same meaning whether it refers to something in the map or the territory.)
But for this strategy you really should give explicit paraphrases or, ideally, a method for paraphrasing all mathematical truths.
Isn’t that exactly what I did? The method I gave is that you break down the truth assertion into a claim about the correctness of the terms (i.e., part 1, what the common usage is), and a claim about whether the model in common usage implies the truth (part 2). I suppose I could have been more specific about how exactly one does that, but it seems straightforward enough that I didn’t think I needed to give more detail.
You’ve erred from the moment you think that that a true statement implies independent existence of its predicates, leaving out the possibility that the predicates exist only as part of models.
No, no. Not the predicates. Just the values of it’s bound variables. I’m not saying “Primeness exists”. See my reply to wedrifid.
Isn’t that exactly what I did? The method I gave is that you break down the truth assertion into a claim about the correctness of the terms (i.e., part 1, what the common usage is), and a claim about whether the model in common usage implies the truth (part 2). I suppose I could have been more specific about how exactly one does that, but it seems straightforward enough that I didn’t think I needed to give more detail.
So “3 is prime” means what? “2+3= 5″ means what? You absolutely need to give more detail because as far as I know none of the many very smart people who have tried have been able to give a satisfactory paraphrase for all true mathematical statements.
No, no. Not the predicates. Just the values of it’s bound variables. I’m not saying “Primeness exists”.
How does “3 is prime” imply that “3″ exists, while “primeness is related to the zeroes of the Zeta function” not imply that “primeness” exists?
This whole discussion is, ultimately, about the definition of the word “exist”. But if you try to hang that definition off linguistic phenomena, then you’re at the whim of every linguistic construct people can come up with, and people can probably twist words to get something covered by your definition that you didn’t want to be.
How does “3 is prime” imply that “3″ exists, while “primeness is membership in the set of zeroes of the Zeta function” not imply that “primeness” exists?
The latter does imply primeness exists. But “3 is prime” doesn’t. Luckily you haven’t just used primeness as the value of a bound variable, you’ve given an appropriate paraphrase (although now you’re committed to the existence of the set of zeros of the Zeta function).
This whole discussion is, ultimately, about the definition of the word “exist”. But if you try to hang that definition off linguistic phenomena, then you’re at the whim of every linguistic construct people can come up with, and people can probably twist words to get something covered by your definition that you didn’t want to be.
So “3 is prime” means what? “2+3= 5″ means what? You absolutely need to give more detail because as far as I know none of the many very smart people who have tried have been able to give a satisfactory paraphrase for all true mathematical statements.
My method would be to find the associated #1 and #2 statements. The #1 statement would be a claim about what people use the terms “3”, “is”, and “prime” to mean. Under this method you would next identify the common conception of “3″ (by empirical examination of how people use the term and under constraint of Occam’s Razor) as something like, “the quantity immediately following the quantity immediately following the quantity immediately following the quantity of nothing”. Then do the same for the other parts.
(Also, keep in mind that this method is only necessary for the bare statement that “3 is prime”. You needn’t construct the associated #1 statement for more specific claims like, “Here is a system of math. Under those rules and definitions, 3 is prime.”)
Then you would construct the #2 statement, which would be that, under those meanings, the claim as a whole follows from the definitions and assumptions of the system implictly used by those meanings. This would be something like, “under any physical system behaving isomorphically to the assumptions in #1, the physical correlate of ‘3 being prime’ will hold”, and that physical correlate will be something like, “any division of the units correlating to 3 will be such that each partition will have a different number of units, or one unit, or three units”.
...Er, okay, perhaps more detail was needed. Does that answer your question, though?
Yes. That is helpful. I was going to bring this up in my first comment but decided to focus on one thing at a time. Your view seems very similar to nominalist-structuralism, which I find appealing as well. At least I can read nominalist-structuralism into your post and comments. Afaict, it’s considered the most promising version of nominalism going. They basically take your use of isomorphism and go one step farther. The SEP discusses it some but it’s a pretty poorly written article. You might have to do a lot of googling. Structuralism argues that math does not describe objects of any kind but rather, structures and places within structures (and have no identity or features outside those structures). Any given integer is not an object but a places in the structure of integers. As you can imagine once you think of mathematical truths this way it become obvious how math can be used to describe other physical systems: namely those systems instantiate the same structure (or in your words, are isomorphic?). Nominalist structuralism involves disclaiming the existence of structures (which are abstract objects) as independent of the systems that instantiate them
ETA: One issue here is the work being done by the word “resembles” when we say “the structure of real numbers resembles the structure of space’, or “the structure of macroscopic object motion resembles the structure of simulated object motion in your physics simulator”. Which is the same issue as “what is the work being done by ‘isomorphic’.
Voted up because this is a great topic that I’d like us to try and begin to tackle.
But this post really frustrating to try to respond to. Not because it is especially wrong-headed or poorly written but just because it is a little hard for me to find my way around your theory. It is difficult to find a point of traction. In general, I suspect it just isn’t really solving problems but eliding distinctions and ignoring problems (just based on what I do know and the relative shortness of this compared to most other work in philosophy of math). This is pretty much the way I feel about what Eliezer has said on the subject and just about every single thought I’ve ever had on the subject. I’m also not sure I’m familiar enough with the subject area to be able to examine this post in the way it requires.
So I suspect I’ll end up prodding you in a couple places but to begin with: what exactly do you take the Platonist thesis to be? If there is an analogical relationship between a particular expression in our system of inscriptions and our rules for manipulating them (i.e. a written equation) and a physical system (i.e. a system that equation describes) that seems to suggest an underlying structure which is instantiated in both the mathematical expression and the physical system. That such structures exist independently of the mind strikes me as a platonist position. What exactly is wrong with that position? Or what even did you say to contradict it?
Perhaps we need to have a discussion about abstract objects in general before tackling the math.
I do think you’re right about the map-territory confusions here. They definitely abound.
This is not a good heuristic, because in philosophy, works tend to be longest when they’re confused, because most of the length tends to be spent repairing the damage caused by a mistake early on.
So philosophy can get long because the author is running damage control. True. But it can also be short because the author is trying to answer 5-6 questions at once without engaging with the arguments of those who argue against his position. So length by itself- maybe a bad heuristic. But I’m leveraging this heuristic with enough background to make it work.
That there is an immaterial realm of ideal forms (structures, concepts) of which our universe consists solely of imperfect approximations of.
I would say instead that there is some generating function for reality. A system of inscriptions/rules can describe that generating function imperfectly; but this in no way means that the rule/inscription system has some existence apart from its instantiation as the universe itself, and again explicitly in a model.
This stuff about imperfect approximations is just a remnant of Plato’s mysticism. Few modern platonists would say anything like that. This notion of an immaterial “realm” has similar connotations. How about:
Platonism is appealing because it adheres to our norm of accepting the existence of things we make true statements about. “Silas is cool” implies the existence of Silas. Similarly, “3 is prime” implies the existence of 3. The list of non-platonist options as far as I can recall consists of: mathematical objects are mental objects, mathematical objects are physical objects, statements about mathematical objects are false (like statements about Santa Claus), or statements about mathematical objects are actually paraphrases of sentences that don’t commit us to the existence of abstract objects.
It seems like you are trying something like the last. But for this strategy you really should give explicit paraphrases or, ideally, a method for paraphrasing all mathematical truths.
But then what kind of thing is this function? It clearly isn’t merely a set of inscriptions and rules for manipulating them (the models). Nor is it merely the physical universe. We talk like it exists. If it doesn’t, why do we talk like this and what do claims about it really mean?
At least for geometrical forms, the abstractions may be intrinsic to the mind, even if they don’t exist outside it.
In The Man Who Mistook His Wife for a Hat, there’s a description of a man who lost the ability to visually recognize ordinary objects, though he could still see. The one description suggest that he just saw geometry.
In Crashing Through, which is about a man who lost his sight at age 3 and recovered it in middle age and which has a lot about recovered vision and the amount of processing it takes to make sense of what you see, there’s mention of some people who are very disappointed when they recover their sight—they’re constantly comparing the world to an idea of it which is perfectly clean and geometrical.
I’m a little confused: did is visual field lose focus such that, instead of seeing the details on objects and their imperfections he actually just saw idealized geometric figures?
One problem with this as evidence of the possibility that geometric forms could exist only in the human mind is that it presumably only applies to a rather narrow class of geometric forms. It would be weird if the geometric forms we have innate access to had a different ontological status from forms that can’t be instantiated in the human mind: like a 1000-sided polygon or something in 4+ dimensions.
What I meant was that, if people have simple geometric forms built deep into their minds, then it would be tempting to conclude that math has an objective eternal existence because it feels that way.
In any case, I found the actual quote, and I’ve very uncertain that it suggests what I thought it did. It seems as though the man was at least as sensitive to simple topology as geometry, However, people don’t romanticize topology.
Here’s the passage, which I had not remembered as well as I thought:
It’s a wonderful extract in any case. It is fascinating to see someone describing the world without anything more than the phenomenology of his surroundings. It is interesting that the concepts he had access to were mathematical and geometric- that these concepts involve a part of the brain separate from the part that involves more complex and obviously learned concepts like shoe, glove, and flower does seem important to keep in mind when evaluating the evidence on this issue. You’re right that this fact could lead to us positing a false ontological difference… though of course there are those who will say “gloveness” and “flowerness” are abstract objects as well. The fact that these concepts are processed in different parts of the brain could also be taken as evidence for the distinction in that different evolutionary processes generated these two kinds of concepts. I’m not sure how to interpret this. Good for keeping in mind though.
Googling it looks like maybe he just had visual agnosia? Which doesn’t really entail what you’re saying. That would mean that he could see normally but just couldn’t recognize figures as objects with names and functions. Or are you saying the details of objects disappeared and all that was left were the basic geometric forms?
On problem with this as evidence of the possibility that geometric forms could exist only in the human mind is that it presumably only applies to a rather narrow class of geometric forms. It would be weird if the geometric forms we have innate access to had a different ontological status from forms that can’t be instantiated in the human mind: like a 1000-sided polygon or something in 4+ dimensions.
This is very clarifying.
To bring the comparison closer to the mark: It would also imply the existence of ‘cool’.
Heh. What I had in mind was Quine’s criterion for ontological commitment under which it wouldn’t. So Silas is cool is something like, where cool is the predicate letter C: ∃x(Cx ∩ x=”Silas”). We’re committed to the existence of the bound variables (to exist is to be the value of a bound variable) but not of the properties, there doesn’t have to be anything like coolness (assuming that was what you were suggesting).
There is an older argument that claims all words must refer to things and thus a word like “cool” must refer to coolness. But I wasn’t intending to make that argument (though I didn’t say nearly enough in my previous comment to expect everyone to figure that out).
My reading of Silas’s essay (and in particular looking at his diagrams) gave me impression that his ‘2’ is closer to what you would describe as a ‘property’ than the category in which you put ‘Silas’.
I was just starting from the observation that in our mathematical discourse we treat numbers like objects, not properties. “The number between 2 and 4”, “there is a prime number greater than one million”, “5 is odd” etc. all treat numbers as objects.
I would call those properties that had properties. But I’m a programmer, not a mathematician or philosopher (so don’t know which limitations I’m supposed to have placed around my thinking!)
By the way, I think ‘cool’ is kinda ‘lame’ but ‘awesomeness’ is kinda ‘cool’. Just sayin’.
This, I claim, is where you should stop the chain. You’ve erred from the moment you think that that a true statement implies independent existence of its predicates, leaving out the possibility that the predicates exist only as part of models. (Or, equivalently, from treating the term “existence” as having the same meaning whether it refers to something in the map or the territory.)
Isn’t that exactly what I did? The method I gave is that you break down the truth assertion into a claim about the correctness of the terms (i.e., part 1, what the common usage is), and a claim about whether the model in common usage implies the truth (part 2). I suppose I could have been more specific about how exactly one does that, but it seems straightforward enough that I didn’t think I needed to give more detail.
No, no. Not the predicates. Just the values of it’s bound variables. I’m not saying “Primeness exists”. See my reply to wedrifid.
So “3 is prime” means what? “2+3= 5″ means what? You absolutely need to give more detail because as far as I know none of the many very smart people who have tried have been able to give a satisfactory paraphrase for all true mathematical statements.
How does “3 is prime” imply that “3″ exists, while “primeness is related to the zeroes of the Zeta function” not imply that “primeness” exists?
This whole discussion is, ultimately, about the definition of the word “exist”. But if you try to hang that definition off linguistic phenomena, then you’re at the whim of every linguistic construct people can come up with, and people can probably twist words to get something covered by your definition that you didn’t want to be.
The latter does imply primeness exists. But “3 is prime” doesn’t. Luckily you haven’t just used primeness as the value of a bound variable, you’ve given an appropriate paraphrase (although now you’re committed to the existence of the set of zeros of the Zeta function).
Huh?
My method would be to find the associated #1 and #2 statements. The #1 statement would be a claim about what people use the terms “3”, “is”, and “prime” to mean. Under this method you would next identify the common conception of “3″ (by empirical examination of how people use the term and under constraint of Occam’s Razor) as something like, “the quantity immediately following the quantity immediately following the quantity immediately following the quantity of nothing”. Then do the same for the other parts.
(Also, keep in mind that this method is only necessary for the bare statement that “3 is prime”. You needn’t construct the associated #1 statement for more specific claims like, “Here is a system of math. Under those rules and definitions, 3 is prime.”)
Then you would construct the #2 statement, which would be that, under those meanings, the claim as a whole follows from the definitions and assumptions of the system implictly used by those meanings. This would be something like, “under any physical system behaving isomorphically to the assumptions in #1, the physical correlate of ‘3 being prime’ will hold”, and that physical correlate will be something like, “any division of the units correlating to 3 will be such that each partition will have a different number of units, or one unit, or three units”.
...Er, okay, perhaps more detail was needed. Does that answer your question, though?
Yes. That is helpful. I was going to bring this up in my first comment but decided to focus on one thing at a time. Your view seems very similar to nominalist-structuralism, which I find appealing as well. At least I can read nominalist-structuralism into your post and comments. Afaict, it’s considered the most promising version of nominalism going. They basically take your use of isomorphism and go one step farther. The SEP discusses it some but it’s a pretty poorly written article. You might have to do a lot of googling. Structuralism argues that math does not describe objects of any kind but rather, structures and places within structures (and have no identity or features outside those structures). Any given integer is not an object but a places in the structure of integers. As you can imagine once you think of mathematical truths this way it become obvious how math can be used to describe other physical systems: namely those systems instantiate the same structure (or in your words, are isomorphic?). Nominalist structuralism involves disclaiming the existence of structures (which are abstract objects) as independent of the systems that instantiate them
ETA: One issue here is the work being done by the word “resembles” when we say “the structure of real numbers resembles the structure of space’, or “the structure of macroscopic object motion resembles the structure of simulated object motion in your physics simulator”. Which is the same issue as “what is the work being done by ‘isomorphic’.