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Fictionalism in the Philosophy of Mathematics [pdf]

41 points by monort 11y ago

Houshalter 11y ago
This is absurd. Clearly mathematical statements are highly predictive of the real world. They aren't false. These people seem confused over the meaning of the word "exist". Regardless of whether or not numbers "exist", we can show that objects in the real world can obey the same laws as abstract numbers. If I have 2 apples, and take 2 more apples, I will never have 5 apples. The properties of math are real and apply to the real world. If you insist on modelling philosophy on the language we happen to use, then just treat numbers as adjectives. As if 5 is a property an object can have, rather than a physical object itself. You don't need to worry about 2 "existing" any more than worrying about "tallness" exists, when talking about objects that are taller than other objects.
- abrezas 11y ago
  You think that they are confusing the term exists but the reality is that what "exists" means is very hard to define. Where is the line between abstract objects and the real world you mentioned? Can you give a rule that separates the two? Do black holes exist? Electrons? Magnetic fields? Is one of these a model but doesn't really exist?
  - Houshalter 11y ago
    Arguing about the definition of a word is a huge red flag that a discussion is going nowhere. Sadly a lot of philosophy seems to be basically arguing about words. >Do black holes exist? Electrons? Magnetic fields? Yes. I mean we might not know what they are exactly or how they work, but they are clearly real phenomena that we can observe. In the same sense, "twoness" is a real property that a set of objects can have.
wyager 11y ago
There's a bit of a litmus test you can apply to belief systems to find out if they're definitely objective. An intelligent alien species that's never met humans would almost certainly invent math. The syntax and organization would probably be different, but the rules would be the same. On the other hand, aliens would almost certainly not write The Hobbit.
- ezequiel-garzon 11y ago
  As Bertrand Russell put it, mathematics lies "remote even from the pitiful facts of nature". http://www.gutenberg.org/ebooks/25447 http://www.gutenberg.org/ebooks/25447
- calibraxis 11y ago
  I don't know, it may be the current intellectual culture's speciesism that we often think that. For instance, they may have many modes of perception better than math. Or even if they have a "math" sense that's oddly like ours, they may be unaware of it because it's unconscious... in their underlying circuitry... just like we're not built to directly introspect many of our own capacities. We may even be in constant contact with "aliens" now, but unable to perceive them meaningfully. Like most lifeforms maybe couldn't tell us apart from a rock or rubber.
catnaroek 11y ago
> Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at face value and, thus interpreted, are false. Arrrgh! So annoying! How much time will it pass until philosophers of mathematics finally understand that mathematical truth has nothing to do with philosophical truth? > Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Crash course in logic: If mathematical objects don't exist, then statements about them aren't “false” - they're meaningless.
- hyperpape 11y ago
  Don't say things like "crash course in logic", and follow them up with something that is highly debatable. You should read Russell on definite descriptions, (or if you have read him, you should write as if you had). As for your distinction between philosophical truth, and mathematical truth, you should elaborate, because it doesn't sound like any distinction I've ever heard of.
  - catnaroek 11y ago
    > You should read Russell on definite descriptions An expression is only meaningful in a context in which its free variables are bound. So, in a context in which the concept of “prime number” doesn't exist, any statement about “prime numbers” is meaningless. > As for your distinction between philosophical truth, and mathematical truth, you should elaborate Philosophical truth is the correspondence between a statement and reality. Mathematical truth, at least to a formalist like me, is the syntactic property of being substitutable with the symbol “true” in a given context.
    - hyperpape 11y ago
      Philosophers of math aren't confused by this. What they are asking is whether "mathematical truth" has any connection with truth. Formalists more or less deny this, Platonists affirm it. As a side note, correspondence with reality is a contentious definition of truth. It's just one theory of what truth is. I don't really know what to say about your claims about meaninglessness, except that I think they're false. But I don't even know where to start in arguing that. And while I don't think this is much of an argument, I don't think there are really any philosophers, mathematicians, or logicians who have worked out any theory along those lines.
__mbm__ 11y ago
It is interesting to interpret the body of mathematics as you would a collection of fictional tales. However, the philosophy begins to unravel (to me) when it asserts that "8 is larger than 5" is false while "Sydney is larger than San Francisco" is true because the latter statement "has referents". What is it that makes Sydney and San Francisco real objects with meaningful sizes while 8 and 5 are not real and do not have meaningful sizes? Sydney and San Francisco are defined by political and legal "stories" in the same way that 8 and 5 are defined in mathematical "stories". The theory only seems to be consistent if all out-of-context falsifiable statements are taken to be false. This theory placates me, since it leaves the truth value of mathematical statements (in the context of the mathematical story) to mathematicians. However, it renders any conclusions meaningless to mathematics, even if it is meaningful for a philosophy dealing with human stories.
- Retra 11y ago
  I've always considered true and false to be properties of models, which are (necessarily) approximations of some underlying reality. So I'd say "8 is larger than 5 in some sense" and "Sydney is larger than San Francisco in some sense", where we might admit that the senses differ, and maybe 8 > 5 is 'truer' because the senses in which it are true are more general and require fewer experiences to verify. But at the end of the day, you don't want to just say "everything can be true if you stretch far enough", you want to say that things are true only when we've demonstrated some utility in saying that they're true. So we just defer the issue: if you want to say something is true, you always need to know what difference it makes.
  - __mbm__ 11y ago
    > if you want to say something is true, you always need to know what difference it makes. Absolutely. It seems that fictionalism avoids both the "in some sense" qualifications and worrying about what difference it makes by asserting that statements out of context are simply false. While consistent, I'm similarly not convinced that its useful.
- hyperpape 11y ago
  You may just be hung up on the examples of Sidney and San Francisco, which are not essential to the point. Suppose I bring you a 5 pound weight a 1 pound weight, and say "this is heavier than that". Then we can substitute those for Sydney and San Francisco in the example. Put another way, the metaphysical status of cities is not a direct consequence or assumption of the metaphysics of mathematical entities.
force_reboot 11y ago
Fictionalism, and other formalist theories, will have to confront that problem that when we speak of mathematics as abstract rules governing strings of symbols, these rules themselves are mathematical. So it only replaces "numbers are real" with "abstract symbols are real". There are axiomatic systems that are strong enough to express manipulations of abstract symbols, but weaker than the usual systems that mathematicians deal with (e.g. see the work of Edward Nelson on so called predicative arithmetic). But to my knowledge these have not been explored much.
jules 11y ago
Does this represent the state of philosophy of mathematics, because I find this view quite naive. Mathematics is in the business of making a model of the real world and then making falsifiable predictions with it, just like science. Take Fermat's last theorem: for all positive a,b,c and n>2, the value a^n + b^n - c^n never comes out 0. This is an empirically falsifiable statement. You can even make it a statement about real world objects if you wish: represent a number by a jar of coins, and do addition C=A+B by filling jar C with the coins in jar A and jar B together, multiplication AB by taking a whole jar B for each coin in A, etc. Rules of mathematical deduction are just devices for making predictions. So mathematical objects are neither "real" as in platonism, nor meaningless as in fictionalism.
- roywiggins 11y ago
  Embodied mathematics is what you are describing I think. The idea is that mathematics arises out of ordinary experience extended by metaphor. It charts a pretty satisfying middle ground between Platonism and Fictionalism. Mathematics is a story, but it is "at the bottom" tied into our experience of everyday reality. Do real numbers exist? Probably not, but the the continuum is like a long stick you can make marks on. Can you have a trillion apples, so you know that a trillion + a trillion is two trillion? No, but you can reason about "trillion" as existing metaphorically.
  - hyperpape 11y ago
    That actually just sounds like good old fashioned nominalism or fictionalism--you'd have to sketch out a more detailed story, but I don't think it's actually distinct.
    - roywiggins 11y ago
      I'm (probably badly) summarizing Lakoff & Nunez's book: https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From Neither are capital-P Philosophers, and I don't remember if they actually make a claim that their ideas are philosophically distinct from fictionalism- they're more worried about platonists! The other approach I'm fond of is saying mathematical objects exist but only inside the heads of mathematicians. As long as mathematicians share a particular idea and can talk about it, it's math. So when I point to the real numbers, I'm pointing at a real thing: the shared idea of them. There's no Platonic realm, but they exist in the mind. It parallels how mathematics is really done (a proof is only a proof if it convinces mathematicians) so I find it compelling on that front.
KKKKkkkk1 11y ago
I'm having trouble understanding the significance of Hartry Field's work. It sounds like he replaced the axiomatic system of what we call mathematics with an axiomatic system that he developed specifically for Newtonian gravity, and based on this he was able to claim that mathematics is dispensable. As an undergrad in logic class, I got the impression that mathematics is the discipline that studies axiomatic systems, so if you build an axiomatic system, you're doing mathematics. If that's true, then isn't all he did is just redefining the word mathematics in a very narrow sense and then dispensing with that narrowly defined notion?
- hyperpape 11y ago
  The question concerns the quantifiers that you see in math ("there exists an even prime number", "all numbers have a unique prime factorization"), etc. Field shows that for the theories necessary for Newtonian mathematics, we can treat those as quantifying over points in space-time, as opposed to anything distinctively mathematical. So we avoid Platonism. It sounds like you, like most people who haven't studied philosophy of math, assume some sort of pseudo-formalist account, and therefore don't find the question very compelling. Please don't take that disparagingly (most mathematicians are in the same boat, and I'm not sure I have much of an opinion at this point, save to note that people who haven't studied philosophy tend to think the issue is obvious in a way that a lot of philosophers don't).
  - samth 11y ago
    I think most mathematicians are more likely to be Platonists than formalists. Formalism is also a real position in philosophy of math, but a minority one for I think two reasons. One is that the math we've developed seems significant enough and effective enough to deserve more explanation than just "it's one of many possible formal systems". Second, if you take a straightforward formalist approach, then there aren't really any interesting questions in philosophy of math, so people who think that are less likely to become philosophers of math.
    - cokernel 11y ago
MaysonL 11y ago
Whatever one chooses to say about the truth or falsity of mathematical statements (& imo, calling them all false is ludicrous), it is hard to refute the argument that they are quite effective. See Hamming and Wigner[0,1] [0]http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.htm... [1]https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...

41 points by monort 11y ago

This is absurd. Clearly mathematical statements are highly predictive of the real world. They aren't false. These people seem confused over the meaning of the word "exist". Regardless of whether or not numbers "exist", we can show that objects in the real world can obey the same laws as abstract numbers. If I have 2 apples, and take 2 more apples, I will never have 5 apples. The properties of math are real and apply to the real world. If you insist on modelling philosophy on the language we happen to use, then just treat numbers as adjectives. As if 5 is a property an object can have, rather than a physical object itself. You don't need to worry about 2 "existing" any more than worrying about "tallness" exists, when talking about objects that are taller than other objects.

abrezas 11y ago
You think that they are confusing the term exists but the reality is that what "exists" means is very hard to define. Where is the line between abstract objects and the real world you mentioned? Can you give a rule that separates the two? Do black holes exist? Electrons? Magnetic fields? Is one of these a model but doesn't really exist?
- Houshalter 11y ago
  Arguing about the definition of a word is a huge red flag that a discussion is going nowhere. Sadly a lot of philosophy seems to be basically arguing about words. >Do black holes exist? Electrons? Magnetic fields? Yes. I mean we might not know what they are exactly or how they work, but they are clearly real phenomena that we can observe. In the same sense, "twoness" is a real property that a set of objects can have.

wyager 11y ago

There's a bit of a litmus test you can apply to belief systems to find out if they're definitely objective. An intelligent alien species that's never met humans would almost certainly invent math. The syntax and organization would probably be different, but the rules would be the same. On the other hand, aliens would almost certainly not write The Hobbit.

ezequiel-garzon 11y ago
As Bertrand Russell put it, mathematics lies "remote even from the pitiful facts of nature". http://www.gutenberg.org/ebooks/25447 http://www.gutenberg.org/ebooks/25447
calibraxis 11y ago
I don't know, it may be the current intellectual culture's speciesism that we often think that. For instance, they may have many modes of perception better than math. Or even if they have a "math" sense that's oddly like ours, they may be unaware of it because it's unconscious... in their underlying circuitry... just like we're not built to directly introspect many of our own capacities. We may even be in constant contact with "aliens" now, but unable to perceive them meaningfully. Like most lifeforms maybe couldn't tell us apart from a rock or rubber.

catnaroek 11y ago

> Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at face value and, thus interpreted, are false. Arrrgh! So annoying! How much time will it pass until philosophers of mathematics finally understand that mathematical truth has nothing to do with philosophical truth? > Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Crash course in logic: If mathematical objects don't exist, then statements about them aren't “false” - they're meaningless.

hyperpape 11y ago
Don't say things like "crash course in logic", and follow them up with something that is highly debatable. You should read Russell on definite descriptions, (or if you have read him, you should write as if you had). As for your distinction between philosophical truth, and mathematical truth, you should elaborate, because it doesn't sound like any distinction I've ever heard of.
- catnaroek 11y ago
  > You should read Russell on definite descriptions An expression is only meaningful in a context in which its free variables are bound. So, in a context in which the concept of “prime number” doesn't exist, any statement about “prime numbers” is meaningless. > As for your distinction between philosophical truth, and mathematical truth, you should elaborate Philosophical truth is the correspondence between a statement and reality. Mathematical truth, at least to a formalist like me, is the syntactic property of being substitutable with the symbol “true” in a given context.
  - hyperpape 11y ago
    Philosophers of math aren't confused by this. What they are asking is whether "mathematical truth" has any connection with truth. Formalists more or less deny this, Platonists affirm it. As a side note, correspondence with reality is a contentious definition of truth. It's just one theory of what truth is. I don't really know what to say about your claims about meaninglessness, except that I think they're false. But I don't even know where to start in arguing that. And while I don't think this is much of an argument, I don't think there are really any philosophers, mathematicians, or logicians who have worked out any theory along those lines.

__mbm__ 11y ago

It is interesting to interpret the body of mathematics as you would a collection of fictional tales. However, the philosophy begins to unravel (to me) when it asserts that "8 is larger than 5" is false while "Sydney is larger than San Francisco" is true because the latter statement "has referents". What is it that makes Sydney and San Francisco real objects with meaningful sizes while 8 and 5 are not real and do not have meaningful sizes? Sydney and San Francisco are defined by political and legal "stories" in the same way that 8 and 5 are defined in mathematical "stories". The theory only seems to be consistent if all out-of-context falsifiable statements are taken to be false. This theory placates me, since it leaves the truth value of mathematical statements (in the context of the mathematical story) to mathematicians. However, it renders any conclusions meaningless to mathematics, even if it is meaningful for a philosophy dealing with human stories.

Retra 11y ago
I've always considered true and false to be properties of models, which are (necessarily) approximations of some underlying reality. So I'd say "8 is larger than 5 in some sense" and "Sydney is larger than San Francisco in some sense", where we might admit that the senses differ, and maybe 8 > 5 is 'truer' because the senses in which it are true are more general and require fewer experiences to verify. But at the end of the day, you don't want to just say "everything can be true if you stretch far enough", you want to say that things are true only when we've demonstrated some utility in saying that they're true. So we just defer the issue: if you want to say something is true, you always need to know what difference it makes.
- __mbm__ 11y ago
  > if you want to say something is true, you always need to know what difference it makes. Absolutely. It seems that fictionalism avoids both the "in some sense" qualifications and worrying about what difference it makes by asserting that statements out of context are simply false. While consistent, I'm similarly not convinced that its useful.
hyperpape 11y ago
You may just be hung up on the examples of Sidney and San Francisco, which are not essential to the point. Suppose I bring you a 5 pound weight a 1 pound weight, and say "this is heavier than that". Then we can substitute those for Sydney and San Francisco in the example. Put another way, the metaphysical status of cities is not a direct consequence or assumption of the metaphysics of mathematical entities.

force_reboot 11y ago

Fictionalism, and other formalist theories, will have to confront that problem that when we speak of mathematics as abstract rules governing strings of symbols, these rules themselves are mathematical. So it only replaces "numbers are real" with "abstract symbols are real". There are axiomatic systems that are strong enough to express manipulations of abstract symbols, but weaker than the usual systems that mathematicians deal with (e.g. see the work of Edward Nelson on so called predicative arithmetic). But to my knowledge these have not been explored much.

jules 11y ago

Does this represent the state of philosophy of mathematics, because I find this view quite naive. Mathematics is in the business of making a model of the real world and then making falsifiable predictions with it, just like science. Take Fermat's last theorem: for all positive a,b,c and n>2, the value a^n + b^n - c^n never comes out 0. This is an empirically falsifiable statement. You can even make it a statement about real world objects if you wish: represent a number by a jar of coins, and do addition C=A+B by filling jar C with the coins in jar A and jar B together, multiplication AB by taking a whole jar B for each coin in A, etc. Rules of mathematical deduction are just devices for making predictions. So mathematical objects are neither "real" as in platonism, nor meaningless as in fictionalism.

roywiggins 11y ago
Embodied mathematics is what you are describing I think. The idea is that mathematics arises out of ordinary experience extended by metaphor. It charts a pretty satisfying middle ground between Platonism and Fictionalism. Mathematics is a story, but it is "at the bottom" tied into our experience of everyday reality. Do real numbers exist? Probably not, but the the continuum is like a long stick you can make marks on. Can you have a trillion apples, so you know that a trillion + a trillion is two trillion? No, but you can reason about "trillion" as existing metaphorically.
- hyperpape 11y ago
  That actually just sounds like good old fashioned nominalism or fictionalism--you'd have to sketch out a more detailed story, but I don't think it's actually distinct.
  - roywiggins 11y ago
    I'm (probably badly) summarizing Lakoff & Nunez's book: https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From Neither are capital-P Philosophers, and I don't remember if they actually make a claim that their ideas are philosophically distinct from fictionalism- they're more worried about platonists! The other approach I'm fond of is saying mathematical objects exist but only inside the heads of mathematicians. As long as mathematicians share a particular idea and can talk about it, it's math. So when I point to the real numbers, I'm pointing at a real thing: the shared idea of them. There's no Platonic realm, but they exist in the mind. It parallels how mathematics is really done (a proof is only a proof if it convinces mathematicians) so I find it compelling on that front.

KKKKkkkk1 11y ago

I'm having trouble understanding the significance of Hartry Field's work. It sounds like he replaced the axiomatic system of what we call mathematics with an axiomatic system that he developed specifically for Newtonian gravity, and based on this he was able to claim that mathematics is dispensable. As an undergrad in logic class, I got the impression that mathematics is the discipline that studies axiomatic systems, so if you build an axiomatic system, you're doing mathematics. If that's true, then isn't all he did is just redefining the word mathematics in a very narrow sense and then dispensing with that narrowly defined notion?

hyperpape 11y ago
The question concerns the quantifiers that you see in math ("there exists an even prime number", "all numbers have a unique prime factorization"), etc. Field shows that for the theories necessary for Newtonian mathematics, we can treat those as quantifying over points in space-time, as opposed to anything distinctively mathematical. So we avoid Platonism. It sounds like you, like most people who haven't studied philosophy of math, assume some sort of pseudo-formalist account, and therefore don't find the question very compelling. Please don't take that disparagingly (most mathematicians are in the same boat, and I'm not sure I have much of an opinion at this point, save to note that people who haven't studied philosophy tend to think the issue is obvious in a way that a lot of philosophers don't).
- samth 11y ago
  I think most mathematicians are more likely to be Platonists than formalists. Formalism is also a real position in philosophy of math, but a minority one for I think two reasons. One is that the math we've developed seems significant enough and effective enough to deserve more explanation than just "it's one of many possible formal systems". Second, if you take a straightforward formalist approach, then there aren't really any interesting questions in philosophy of math, so people who think that are less likely to become philosophers of math.
  - cokernel 11y ago

MaysonL 11y ago

Whatever one chooses to say about the truth or falsity of mathematical statements (& imo, calling them all false is ludicrous), it is hard to refute the argument that they are quite effective. See Hamming and Wigner[0,1] [0]http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.htm... [1]https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...