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.
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.
> 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.
A disclaimer: I am not currently a fictionalist, but I find it remarkably interesting to consider.
Fictionalists are not making claims about mathematical truth; they are making claims about philosophical truth. When Colyvan (who I believe is not a fictionalist) describes fictionalism as an error theory of mathematical discourse, he does not mean that mathematicians incorrectly assign the label of mathematical truth to statements that are mathematically false, but rather that it is an error to conflate mathematical truth ("true in the story of mathematics") with philosophical truth. In other words, you and the fictionalists are in agreement on your first point.
I don't quite understand your second point, or its justification. For example, would you consider the statement "All even primes greater than 2 are divisible by 1001" to be true, false, or meaningless? I think most people will agree that even primes greater than 2 do not exist. Fictionalists would assert that this statement is false; with my background in formal logic, I tend to believe that this statement is vacuously true. I would be interested in seeing an argument for the claim that this statement is meaningless.
The argument of the paper (which I do not find compelling) seems to be that the sentence "all even primes greater than 2 are divisible by 1001" is 'false' because "the words prime, 2, divisible, and 1001" are undefined.
(In the framework in which those words aren't undefined, then sure, we can talk about whether that sentence is true or false and have an interesting discussion. In formal logic it's definitely true. But that's not the discussion the article is interested in, as far as I can tell.)
To which OP (and I, and I think most people..) would say: we accept that you're being clever by declaring that "outside of mathematics, these words are undefined". But calling a sentence with undefined terms "false" is absurd. Just call it "meaningless" until all the terms are defined.
Then you avoid making statements like "'8+5=13' is false", which at first glance seems absurd, and at second glance seems like "you're saying things that are absurd to raise eyebrows instead of taking the easy alternative of saying things that are reasonable."
The Wikipedia link is the reference I intended. "Captain Obvious" refers to the one in a group or conversation who states what everyone else knows, and as such impedes the group's progression toward their shared goals.
Thanks, HN user "douche", you nailed it.
Strange to me how folk wisdom is so unwelcome in a philosophy discussion. While not explicitly stated, the subtext appears to be putting my comment as the "Captain Obvious." I'll avoid commenting in this context in the future.
> mathematical truth has nothing to do with philosophical truth
This is not entirely true. Philosophy of math has a lot of influence on mathematical truth. Intuitionism, for example, is a strong one. Intuitionism's influence isn't just in mathematical truth, but also in programming languages.
While I appreciate very much the technical contributions of intuitionists, such as logics without excluded middle, and computational interpretations of inference rules, I just can't bring myself to accept the idea that the truth of mathematical statements is conditional on our ability to validate them using our intuition. To me, intuition is a crutch that I use to overcome my inability to accurately perform long calculations. But the ultimate arbiter of truth is the collection of rules of the formal system in question.
This is probably naive, but is it not the case that mathematical truths, and also mathematical falses, are 'just' the provable implications of your chosen axioms? The search for philosophical truth seems to lead inexorably to the point where you are trying to persuade your peers to accept your definition of words, which perhaps can be rephrased as persuading them to accept your axioms?
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.
> 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.
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.
> I don't really know what to say about your claims about meaninglessness, except that I think they're false.
Syntax has no meaning outside of the rules for manipulating it. If no rule applies to a particular expression, then it has no meaning whatsoever. And, in any sensible formal system, like typed lambda calculi with Church-style semantics, expressions only have meaning in contexts where their free variables are bound.
It sounds like you're thinking of a statement like "there is a prime number" being interpreted as Fa (where a purports to be the name of a number, and F means exist). But really, it should be interpreted as Ex Fx--there exists some x such that Fx.
The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it.
This is much like the mistake that Russell criticized in On Denoting, with the difference that he discussed definite descriptions. But the error is the same: he attacked opponents who thought that "the present king of france is bald" was either meaningless, or required some sort of bizzaro existence for the present King of France. And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France.
> It sounds like you're thinking of a statement like "there is a prime number" being interpreted as Fa (where a purports to be the name of a number, and F means exist). But really, it should be interpreted as Ex Fx--there exists some x such that Fx.
This is a very low insult - to assume I don't understand the distinction between “F(a)”, where “a” appears free, and “Ex. F(x)”, where “x” appears bound. I understand logical quantifiers and variable binding just fine, thanks.
> The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it.
Another very low insult - to assume that I would conflate “false” with “meaningless”. “There is a natural number that is both even and odd” is false. “There is a myppit inside the quxxit” is meaningless unless we first define: (0) what a “myppit” is, (1) what a “quxxit” is, (2) what specific “quxxit” we're talking about.
The one who's making a mistake is you, because “F” isn't the same as “F(x)”, but rather “λx.F(x)”. In a context where the variable “x” isn't bound, the the latter is meaningful, but the former is not. In other words, the judgment “Γ ⊢ F(x) : Prop”, asserting that “F(x)” is indeed a proposition in the context “Γ”, shouldn't be derivable if “Γ” doesn't contain “x”. But the judgment “Γ ⊢ λx.F(x) : T → Prop”, asserting that “F” is a predicate ranging over “T”s, is derivable from “Γ, x:T ⊢ F(x) : Prop”, and “Γ, x:T” is a context containing “x”.
> And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France.
Unfortunately, “Bald(king)“ isn't the same as “E!person. King(person) ∧ Bald(person)”.
Regarding insults, you started by suggesting that philosophers of math were ignorant of the distinction between truth as either proof theoretic derivation from axioms, or truth in a model and truth simpliciter, I don't think you have room to worry about who is being insulting. Philosophers typically would be introduced to these concepts in advanced undergraduate logic or early on in grad school.
As for the rest of it, I'll leave with one final thought: as a meta-philosophical point, we should rarely attribute meaninglessness to statements unless we are forced to. And it's easy enough to show how to interpret existentials as meaningful even in the absence of a referent. You would have to have very compelling reasons for analyzing mathematical statements the way you want to, and I haven't heard any, only your conviction that it's right.
> we should rarely attribute meaninglessness to statements unless we are forced to.
You don't “attribute meaninglessness” to statements. Syntax is a priori meaningless, and the rules for manipulating it give it meaning. “Meaninglessness” is thus just the lack of a given meaning.
> You would have to have very compelling reasons for analyzing mathematical statements the way you want to, and I haven't heard any, only your conviction that it's right.
Here's a good reason: leveraging the power and reliability of computers for doing mathematics, which is only possible if syntax is mechanically interpretable. The kind of word tricks you suggest, like interpreting a sentence containing the phrase “the King of France” as implicitly asserting the existence (and uniqueness) of a King of France (in addition to whatever you explicitly say about him), are incompatible with a nice, compositional, crystal clear, reliable, mechanical interpretation of syntax.
The concept of prime numbers and the prime numbers themselves is a different thing.
Unicorns don't exist but statements about them aren't meaningless. Fictionalism accepts the concept of mathematical entities but denies their existence in the real world.
> The concept of prime numbers and the prime numbers themselves is a different thing.
I'm okay with that, and haven't said anything suggesting otherwise. However, if you want to reject the existence of mathematical objects in the real world, you have to reject the existence of mathematical definitions too, because mathematical definitions are mathematical objects just like any other. So, if the definition of “prime number” doesn't exist in the real world, then any statement about prime numbers is meaningless in the real world.
> Unicorns don't exist but statements about them aren't meaningless.
Unicorns don't exist in the real world, but the definition of “unicorn” does - just grab any dictionary. It's on the basis the existing definition that you can say “unicorns don't exist”. On the other hand, you may reject the definition of “horse”, but, if you don't, you have to accept that horses exist in the real world. And, if you do reject it, then you also have to reject statements about horses as completely meaningless.
> Fictionalism accepts the concept of mathematical entities but denies their existence in the real world.
The question of whether mathematical objects exist in the real world is ill-posed in the first place. It's like asking if love can be stored in boxes.
At some point mathematical objects are defined with meta-mathematical definitions (axioms etc), so we can accept definitions based on those terms.
But in any case. You obviously do not believe that mathematical objects exist in the first place and it the question is ill posed based on your concept of reality and existence.
That's understandable, but the debate is about which is the right definition of existence reality etc. If you are not interested in this discussion and know the answers then you have nothing to gain from this paper or this discussion.
> At some point mathematical objects are defined with meta-mathematical definitions (axioms etc), so we can accept definitions based on those terms.
“Metamathematics” doesn't exist in isolation from the rest of mathematics. If you reject the existence of mathematical objects (in the real world or elsewhere), you must also reject the existence of metamathematical definitions (in the same context), because the former include the latter.
> But in any case. You obviously do not believe that mathematical objects exist in the first place and it the question is ill posed based on your concept of reality and existence.
I only denied the meaningfulness of asking whether mathematical objects exist in the real world. The terms “real world”, “reality”, etc., I prefer to reserve for that which can be apprehended through physical experience. The distinguishing feature of reality is that you can't reject its existence (in the same way I hypothesized rejecting the definition of the word “horse” in my previous comment) - your senses force you to accept it. If someone cuts through your skin with the intention to bleed you to death, you will feel pain. You may resign yourself to your fate, but that won't make the pain go away. On the other hand, if you find it useless, annoying, etc. to do mathematics, all you need to do is stop doing it.
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.
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.
> 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.
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.
So... the assertion is that physical things are real and numbers are not? It's alluring to accept this axiom, but when challenging a platonic view of mathematics, I don't think that we should accept that without discussion.
In other words, I'm supposed to entertain that math is a fictional tale with fanciful characters called "numbers" that don't exist outside of the story, but the boundaries of so-called physical objects are so apparent that they shouldn't be questioned?
Most physical boundaries are arbitrary, part of the stories that we tell ourselves, and not meaningful in a deep sense. I'd like to know how mathematics, and numbers in particular, are different.
[As an aside: Is it possible to convincingly argue that "this is larger than that" without using numbers?]
> It's alluring to accept this axiom, but when challenging a platonic view of mathematics, I don't think that we should accept that without discussion.
Not sure I follow. It's not an axiom, but the conclusion of a bit of argument, so it's not accepted without discussion.
If you mean my statement, then yes, I'm going to assume that rocks are real (as almost everyone except maybe Idealists) do, but not assume mathematical objects are real, for the sake of the present debate.
I may be misunderstanding, but the author asserts that Sydney and San Francisco are real (or rather, they have referents), and I'm confused because, in particular, Sydney and San Francisco are real in essentially the same way that numbers are real. Your weight example is more concrete, but I was trying to make the point that it suffers (at a less apparent level) the same problem.
I don't know what you mean by "essentially the same", but there are some significant differences. There are reasons you might doubt that San Francisco exists, the primary two of which are:
1) you believe that only things described by fundamental physics are real, everything else we talk about is either reducible to physics, or some sort of strictly inaccurate approximation to some physical reality.
2) You don't believe (1), but still think that for some reason San Francisco seems too vague of an object.
But that said, if San Francisco does exist, it exists in space and time. It didn't exist until the past 500 years, though the land it inhabits existed before then. It's also between 1 and 13 thousand miles from the Easter coast of China. You can locate it, you can go to it, etc.
I cannot speak for _mbm_, but he may be saying that while the urban area known as San Francisco exists, the City of San Francisco is a legal, and therefore abstract, entity. I take your point that it has referents, namely the urban area.
If there are five of something in the universe, is that a referent for the number 5?
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.
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.
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.
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.
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.
I pretty much agree with this position, although I find the "fill ... together" and "bring ... together" metaphors silly.
For example, if you have two jars containing 50 plutionium coins each, putting all of them in the same jar will have some unforeseen consequences. However, this does not change the fact that there are 100 coins altogether.
It fails because you apply the model to a situation where it isn't applicable. This isn't very different than applying Newtonian physics with objects moving near the speed of light.
By the way, numbers aren't defined as jars of coins. Numbers are a device to make predictions, and you can apply that device to make predictions about jars of coins.
I think you find it silly because you know too much about numbers and have forgotten the early years when you learned about numbers. When you teach a little kid numbers you don't start with Peano's axioms, you start with jars of coins and then you build an abstract model and you convince the kid that the abstract model makes accurate predictions with a series of (thought) experiments.
An interesting point but I'm not sure that it does. The ultra-finitist says that numbers above a certain limit don't exist. In my view numbers don't "exist" in the first place. The theory of numbers is a tool for making predictions. The statement that numbers above a certain limit don't exist turns into the statement that examples to test the theory don't exist above a certain limit, because you won't be able to find that big of an example in our universe. IMO this is a much more sensible statement. An ultra-finitist may worry about an intermediate number in a calculation not existing, even though the start and end number may be small (for instance consider log_2(2^n) = n). This is silly in my view, since the theory of numbers is a prediction tool as a whole, and there is no need to physically realize intermediate numbers. As long as you're using the rules of the theory you're good (for the same reason you don't need to be afraid of using infinite sets in the middle of a deduction). One might also worry that the theory of numbers may fail to make correct predictions for very large numbers, but this is really a different question about generalising from examples. It could already fail for very large examples that we can test in our universe. The possibility of it failing for numbers larger than we can realize in our universe is precisely what you don't need to worry about.
All mathematical statements in constructive logic have an experimental test (actually since nonconstructive logic can be embedded in constructive logic via a double negation translation, they do too). Of course this isn't always the most interesting test, since usually you are modeling some higher level concept inside the logic, and you would have a more high level test. For example for the calculation of pi you would measure the ratio between the diameter and perimeter of a circle.
Pi isn't defined by how we measure it. It's defined mathematically. Predicting how a measurement of the ratio between the perimeter and the diameter of a circle comes out is just one of the predictions we can make having calculated pi inside mathematics.
But if at some point further decimals cannot be experimentally proven to be right, at best you can say "pi is such and such for such precision" and any definition that gives the same number up to that precision must be accordingly accepted. Otherwise you need to demonstrate an experiment that uses further precision.
But we don't do that, and that's why mathematics don't have to do with experience, they are an entirely different tool that also happens to be useful in experimental science.
"why mathematics don't have to do with experience" is too strong. Nothing you just said implies that math isn't intimately connected to empirical predictions in some way. What it shows is that contrary to Jules' earlier statements, there's no statement by statement correspondence--for any given mathematical statement, you can't find an interesting empirical prediction you associate with it.
I agree with this statement (depending on how you interpret "change": if the world was different we may have been lead to prove different theorems), and nothing I've said previously contradicts it.
Substitute non-trivial for interesting. For instance, a calculation of pi to 50 digits will map to the same test as a calculation of pi to 51 digits. But the claims have different content so they should map to different tests!
If mathematics doesn't have to do with experience, how do you explain the fact that when we measure pi it's 3.14?
Pi is just pi defined in a mathematical way. It's true that there are other numbers or objects which can make a prediction about the measured ratio of the perimeter to the diameter, but those are not pi. Whether those other objects may be equally valid as pi for making that prediction depends on the details. There may be reasons to prefer one to the other even if they make the same predictions up to measurement precision. We usually prefer the simpler explanation for example. This is equally true in physics and other subjects. Note that as a device for predicting the ratio of the perimeter to the diameter, pi is not perfect. Our space is curved, so for large circles the ratio will deviate from pi, and you have to use a more complicated method based on Riemannian manifolds.
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?
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).
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.
It's possible that a mathematician with philosophical commitments is more likely to be a Platonist than a formalist, but I suspect that most mathematicians are neither ("Philosophy is for the philosophers"). The so-called "naive" view that most mathematicians may be described as having ("I'm discovering something real when I prove a theorem", "Mathematics is objective", and so on) certainly sounds similar to the language used by Platonists, but I have yet to run into a mathematician who is comfortable with the logical conclusions of a strict Platonism. In particular, there are epistemological problems with Platonism (how can we know anything about non-physical entities?) that cause problems in the justification of mathematics and would be nice to be able to ignore.
I think we mostly agree on the empirical issues -- mathematicians usually don't care about the philosophical issues, and espouse naive Platonism which they haven't thought through.
However, I don't think it's right to say that they're formalist just because they wouldn't agree, upon reflection, with Platonism's conclusions. I (like a lot of computer scientists) am a formalist, but it's not fair to say that other people secretly agree with me because I think Platonism has major flaws.
I'm just having trouble understanding what is the difference between the points of a suitable geometrical space and the points of SPACETIME (other than the suggestive name). I guess that this difference is key to philosophers while most mathematicians and physicists would fail to grasp it.
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.
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.
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]
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[ 2.8 ms ] story [ 48.2 ms ] threadAn 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.
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.
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.
Fictionalists are not making claims about mathematical truth; they are making claims about philosophical truth. When Colyvan (who I believe is not a fictionalist) describes fictionalism as an error theory of mathematical discourse, he does not mean that mathematicians incorrectly assign the label of mathematical truth to statements that are mathematically false, but rather that it is an error to conflate mathematical truth ("true in the story of mathematics") with philosophical truth. In other words, you and the fictionalists are in agreement on your first point.
I don't quite understand your second point, or its justification. For example, would you consider the statement "All even primes greater than 2 are divisible by 1001" to be true, false, or meaningless? I think most people will agree that even primes greater than 2 do not exist. Fictionalists would assert that this statement is false; with my background in formal logic, I tend to believe that this statement is vacuously true. I would be interested in seeing an argument for the claim that this statement is meaningless.
(In the framework in which those words aren't undefined, then sure, we can talk about whether that sentence is true or false and have an interesting discussion. In formal logic it's definitely true. But that's not the discussion the article is interested in, as far as I can tell.)
To which OP (and I, and I think most people..) would say: we accept that you're being clever by declaring that "outside of mathematics, these words are undefined". But calling a sentence with undefined terms "false" is absurd. Just call it "meaningless" until all the terms are defined.
Then you avoid making statements like "'8+5=13' is false", which at first glance seems absurd, and at second glance seems like "you're saying things that are absurd to raise eyebrows instead of taking the easy alternative of saying things that are reasonable."
Double negative--I'm not a mathematician, I paint, but Mom taught me right and wrong, two of the latter do not make a right.
"All even primes greater than 2 are divisible by 1001." => "everything else in this analysis will be ignored"
(because it's not math)
Thanks, HN user "douche", you nailed it.
Strange to me how folk wisdom is so unwelcome in a philosophy discussion. While not explicitly stated, the subtext appears to be putting my comment as the "Captain Obvious." I'll avoid commenting in this context in the future.
This is not entirely true. Philosophy of math has a lot of influence on mathematical truth. Intuitionism, for example, is a strong one. Intuitionism's influence isn't just in mathematical truth, but also in programming languages.
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.
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.
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.
Syntax has no meaning outside of the rules for manipulating it. If no rule applies to a particular expression, then it has no meaning whatsoever. And, in any sensible formal system, like typed lambda calculi with Church-style semantics, expressions only have meaning in contexts where their free variables are bound.
The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it.
This is much like the mistake that Russell criticized in On Denoting, with the difference that he discussed definite descriptions. But the error is the same: he attacked opponents who thought that "the present king of france is bald" was either meaningless, or required some sort of bizzaro existence for the present King of France. And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France.
This is a very low insult - to assume I don't understand the distinction between “F(a)”, where “a” appears free, and “Ex. F(x)”, where “x” appears bound. I understand logical quantifiers and variable binding just fine, thanks.
> The second statement is meaningful as long as F is meaningful, and whether F is meaningful is not a matter of whether there are any things such that F. A predicate can be semantically meaningful without having anything satisfy it.
Another very low insult - to assume that I would conflate “false” with “meaningless”. “There is a natural number that is both even and odd” is false. “There is a myppit inside the quxxit” is meaningless unless we first define: (0) what a “myppit” is, (1) what a “quxxit” is, (2) what specific “quxxit” we're talking about.
The one who's making a mistake is you, because “F” isn't the same as “F(x)”, but rather “λx.F(x)”. In a context where the variable “x” isn't bound, the the latter is meaningful, but the former is not. In other words, the judgment “Γ ⊢ F(x) : Prop”, asserting that “F(x)” is indeed a proposition in the context “Γ”, shouldn't be derivable if “Γ” doesn't contain “x”. But the judgment “Γ ⊢ λx.F(x) : T → Prop”, asserting that “F” is a predicate ranging over “T”s, is derivable from “Γ, x:T ⊢ F(x) : Prop”, and “Γ, x:T” is a context containing “x”.
> And Russell's move was to analyze the statement as claiming there is something that is the King of France, there is only one thing that is the King of France, and that thing is bald. And clearly, that is a meaningful conjunction, though there is no present King of France.
Unfortunately, “Bald(king)“ isn't the same as “E!person. King(person) ∧ Bald(person)”.
As for the rest of it, I'll leave with one final thought: as a meta-philosophical point, we should rarely attribute meaninglessness to statements unless we are forced to. And it's easy enough to show how to interpret existentials as meaningful even in the absence of a referent. You would have to have very compelling reasons for analyzing mathematical statements the way you want to, and I haven't heard any, only your conviction that it's right.
You don't “attribute meaninglessness” to statements. Syntax is a priori meaningless, and the rules for manipulating it give it meaning. “Meaninglessness” is thus just the lack of a given meaning.
> You would have to have very compelling reasons for analyzing mathematical statements the way you want to, and I haven't heard any, only your conviction that it's right.
Here's a good reason: leveraging the power and reliability of computers for doing mathematics, which is only possible if syntax is mechanically interpretable. The kind of word tricks you suggest, like interpreting a sentence containing the phrase “the King of France” as implicitly asserting the existence (and uniqueness) of a King of France (in addition to whatever you explicitly say about him), are incompatible with a nice, compositional, crystal clear, reliable, mechanical interpretation of syntax.
Unicorns don't exist but statements about them aren't meaningless. Fictionalism accepts the concept of mathematical entities but denies their existence in the real world.
I'm okay with that, and haven't said anything suggesting otherwise. However, if you want to reject the existence of mathematical objects in the real world, you have to reject the existence of mathematical definitions too, because mathematical definitions are mathematical objects just like any other. So, if the definition of “prime number” doesn't exist in the real world, then any statement about prime numbers is meaningless in the real world.
> Unicorns don't exist but statements about them aren't meaningless.
Unicorns don't exist in the real world, but the definition of “unicorn” does - just grab any dictionary. It's on the basis the existing definition that you can say “unicorns don't exist”. On the other hand, you may reject the definition of “horse”, but, if you don't, you have to accept that horses exist in the real world. And, if you do reject it, then you also have to reject statements about horses as completely meaningless.
> Fictionalism accepts the concept of mathematical entities but denies their existence in the real world.
The question of whether mathematical objects exist in the real world is ill-posed in the first place. It's like asking if love can be stored in boxes.
But in any case. You obviously do not believe that mathematical objects exist in the first place and it the question is ill posed based on your concept of reality and existence.
That's understandable, but the debate is about which is the right definition of existence reality etc. If you are not interested in this discussion and know the answers then you have nothing to gain from this paper or this discussion.
“Metamathematics” doesn't exist in isolation from the rest of mathematics. If you reject the existence of mathematical objects (in the real world or elsewhere), you must also reject the existence of metamathematical definitions (in the same context), because the former include the latter.
> But in any case. You obviously do not believe that mathematical objects exist in the first place and it the question is ill posed based on your concept of reality and existence.
I only denied the meaningfulness of asking whether mathematical objects exist in the real world. The terms “real world”, “reality”, etc., I prefer to reserve for that which can be apprehended through physical experience. The distinguishing feature of reality is that you can't reject its existence (in the same way I hypothesized rejecting the definition of the word “horse” in my previous comment) - your senses force you to accept it. If someone cuts through your skin with the intention to bleed you to death, you will feel pain. You may resign yourself to your fate, but that won't make the pain go away. On the other hand, if you find it useless, annoying, etc. to do mathematics, all you need to do is stop doing it.
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.
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.
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.
Put another way, the metaphysical status of cities is not a direct consequence or assumption of the metaphysics of mathematical entities.
In other words, I'm supposed to entertain that math is a fictional tale with fanciful characters called "numbers" that don't exist outside of the story, but the boundaries of so-called physical objects are so apparent that they shouldn't be questioned?
Most physical boundaries are arbitrary, part of the stories that we tell ourselves, and not meaningful in a deep sense. I'd like to know how mathematics, and numbers in particular, are different.
[As an aside: Is it possible to convincingly argue that "this is larger than that" without using numbers?]
Not sure I follow. It's not an axiom, but the conclusion of a bit of argument, so it's not accepted without discussion.
If you mean my statement, then yes, I'm going to assume that rocks are real (as almost everyone except maybe Idealists) do, but not assume mathematical objects are real, for the sake of the present debate.
But that said, if San Francisco does exist, it exists in space and time. It didn't exist until the past 500 years, though the land it inhabits existed before then. It's also between 1 and 13 thousand miles from the Easter coast of China. You can locate it, you can go to it, etc.
None of those things are true of the number 5 (http://plato.stanford.edu/entries/abstract-objects/).
It's perfectly open for someone to deny that either or both of numbers and San Francisco exists, but the considerations seem a little different.
If there are five of something in the universe, is that a referent for the number 5?
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.
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.
For example, if you have two jars containing 50 plutionium coins each, putting all of them in the same jar will have some unforeseen consequences. However, this does not change the fact that there are 100 coins altogether.
By the way, numbers aren't defined as jars of coins. Numbers are a device to make predictions, and you can apply that device to make predictions about jars of coins.
I think you find it silly because you know too much about numbers and have forgotten the early years when you learned about numbers. When you teach a little kid numbers you don't start with Peano's axioms, you start with jars of coins and then you build an abstract model and you convince the kid that the abstract model makes accurate predictions with a series of (thought) experiments.
But we don't do that, and that's why mathematics don't have to do with experience, they are an entirely different tool that also happens to be useful in experimental science.
> Of course this isn't always the most interesting test
(By the way, they do map to different tests if you view it as a statement in constructive logic, rather than geometry.)
And that's the basic requirement for a semantic theory: map distinct statements to distinct contents.
Pi is just pi defined in a mathematical way. It's true that there are other numbers or objects which can make a prediction about the measured ratio of the perimeter to the diameter, but those are not pi. Whether those other objects may be equally valid as pi for making that prediction depends on the details. There may be reasons to prefer one to the other even if they make the same predictions up to measurement precision. We usually prefer the simpler explanation for example. This is equally true in physics and other subjects. Note that as a device for predicting the ratio of the perimeter to the diameter, pi is not perfect. Our space is curved, so for large circles the ratio will deviate from pi, and you have to use a more complicated method based on Riemannian manifolds.
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).
However, I don't think it's right to say that they're formalist just because they wouldn't agree, upon reflection, with Platonism's conclusions. I (like a lot of computer scientists) am a formalist, but it's not fair to say that other people secretly agree with me because I think Platonism has major flaws.
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.
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?
>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.
[0]http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.htm...
[1]https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...