I’m a little unclear on what your criticism is. Is one of these right?
You’re being too precise, whereas I wanted to have an informal discussion in terms of our everyday intuitions. So definitions are counterproductive; a little unclarity in what we mean is actually helpful for this topic.
There are two kinds of existence, one that holds for Plato’s Realm Of Invisible Mathy Things and one that holds for The Physical World. Your definitions may be true of the Mathy Things, but they aren’t true of things like apples and bumblebees. So you’re committing a category error.
I wanted you to give me a really rich, interesting explanation of what ‘existence’ is, in more fundamental terms. But instead you just copy-pasted a bland uninformative Standard Mathematical Logician Answer from some old textbook. That makes me sad. Please be more interesting next time.
If your point was 1, I’ll want to hear more. If it was 3, then my apologies! If it was 2, then I’ll have to disagree until I hear some argument as to why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)
Thank you, this framework helps. Definitely no to 1. Definitely yes to 2, with some corrections. Yes to some parts of 3.
Re 2. First, let me adopt bounded realism here, with physics (external reality or territory) + logic (human models of reality, or maps). Let me ignore the ultraviolet divergence of decompartmentalization (hence “bounded”), where Many Words, Tegmark IV and modal realism are considered “territory”. To this end, let me put the UV cutoff on logic at the Popper’s boundary: only experimentally falsifiable maps are worth considering. A map is “true” means that it is an accurate representation of the piece of territory it is intended to represent. I apologize in advance if I am inventing new terms for the standard philosophical concepts—feel free to point me to the standard terminology.
Again, “accurate map”, a.k.a. “true map” is a map that has been tested against the territory and found reliable enough to use as a guide for further travels, at least if one does not stray too far. Correspondingly, a piece of territory
is said to “exist” if it is described by an accurate map.
On the other hand, your “invisible mathy things” live in the world of maps. Some of them use the same term “true”, but in a different way: given a set of rules of how to form strings of symbols, true statements are well-formed finite strings. They also use the same term “exist”, but also in a different way: given a set of rules, every well-formed string is said to “exist”.
Now, I am not a mathematician, so this may not be entirely accurate, but the gist is that conflating “exist” as applied to the territory and “exist” as applied to maps is indeed a category error. When someone talks about existence of physical objects and you write out something containing the existential quantifier, you are talking about a different category: not reality, but a subset of maps related to mathematical logic.
I am not sure whether this answers your objection that
why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)
but I hope it makes it clear why I find your replies unconvincing and generally not useful.
You’ve redefined ‘x exists’ to mean ‘x is described by a map that has been tested and so far has seemed reliable to us’, and ‘x is true’ correspondingly. One problem with this is that it’s historical: It commits us to saying ‘Newtonian physics used to be true, but these days it’s false (i.e., not completely reliable as a general theory)‘, and to saying ‘Phlogiston used to exist, but then it stopped existing because someone overturned phlogiston theory’. This is pretty strange.
Another problem is that it’s not clear what it takes to be ‘found reliable enough to use as a guide for further travels’. Surely there’s an important sense in which math is reliable in that sense, hence ‘true’ in the territory-ish sense you outlined above, not just in the map-ish sense. So perhaps we’ll need a more precise definition of territory-ish truth in order to clearly demonstrate why math isn’t in the territory, where the territory is defined by empirical adequacy.
I think your view, or one very close to yours, is actually a lot stronger (can be more easily defended, has broader implications) than your argument for it suggests. You can simply note that things like Abstract Numbers, being causally inert, couldn’t be responsible for the ‘unreasonable efficacy of mathematics’; so that efficacy can’t count as evidence for such Numbers. And nothing else is evidence for Numbers either. So we should conclude, on grounds of parsimony (perhaps fortified with anti-Tegmark’s-MUH arguments), that there are unlikely to be such Numbers. At that point, we can make the pragmatic, merely linguistic decision of saying that mathematicians are using ‘exists’ in a looser, more figurative sense.
Perhaps a few mathematicians are deluded into thinking that ‘exists’ means exactly the same thing in both contexts, but it is more charitable to interpret mathematics in general in the less ontologically committing way, because on the above arguments a platonistic mathematics would be little more than speculative theology. Basically, we end up with a formalist or fictionalist description of math, which I think is very plausible.
You see, we aren’t so different, you and I. Not once we bracket whether unexperienced cucumbers exist out there, anyway!
You’ve redefined ‘x exists’ to mean ‘x is described by a map that has been tested and so far has seemed reliable to us’, and ‘x is true’ correspondingly.
I disagree that this is a redefinition. You believe that elephants exists because you can go and see them, or talk to someone you trust who saw them, etc. You believe that live T-Rex (almost surely) does not exist because it went extinct some 60 odd million years ago. Both beliefs can be updated based on new information.
’Newtonian physics used to be true, but these days it’s false
That’s not at all what I am saying. Consider resisting your tendency to strawman. Newtonian physics is still true in its domain of applicability, it has never been true where it’s not been applicable, though people didn’t know this until 1905.
‘Phlogiston used to exist, but then it stopped existing because someone overturned phlogiston theory’
Again, a belief at the time was that it existed, a more accurate belief (map) superseded the old one and now we know that phlogiston never existed. Maps thought of as being reliable can be found wanting all the time, so the territory they describe is no longer believed to exist, not stopped existing. This is pretty uncontroversial, I would think. Science didn’t kill gnomes and fairies, and such. At least this is the experiment-bounded realist position, as far as I understand it.
You can simply note that things like Abstract Numbers, being causally inert, couldn’t be responsible for the ‘unreasonable efficacy of mathematics’; so that efficacy can’t count as evidence for such Numbers.
I can’t even parse that, sorry. Numbers don’t physically exist because they are ideas, and as such belong in the realm of logic, not physics. (Again, I’m wearing a realist hat here.) I don’t think parsimony is required here. It’s a postulate, not a conclusion.
Perhaps a few mathematicians are deluded into thinking that ‘exists’ means exactly the same thing in both contexts, but it is more charitable to interpret mathematics in general in the less ontologically committing way
Then I don’t understand why you reply to questions of physical existence with some mathematical expressions...
You see, we aren’t so different, you and I. Not once we bracket whether unexperienced cucumbers exist out there, anyway!
I disagree that this is a redefinition. You believe that elephants exists because you can go and see them, or talk to someone you trust who saw them, etc.
Sure, but ‘you believe in X because of Y’ does not as a rule let us conclude ‘X = Y’. I believe in elephants because of how they’ve causally impacted my experience, but I don’t believe that elephants are experiences of mine, or logical constructs out of my experiences and predictions. I believe elephants are animals.
Indeed, a large part of the reason I believe in elephants is that I think elephants would still exist even had you severed the causal links between me and them and I’d never learned about them. The territory doesn’t go away when you stop knowing about it, or even when you stop being able to ever know about it. If you shot an elephant in a rocket out of the observable universe, it wouldn’t stop existing, and I wouldn’t believe it had blinked out of existence or that questions regarding its existence were meaningless, once its future state ceased to be knowable to me.
Elephants don’t live in my map. But they also don’t live in my map-territory relation. Nor do they live in a function from observational data to hypotheses-that-help-us-build-rockets-and-iPhones-and-vaccines. They simply and purely live in the territory.
That’s not at all what I am saying. Consider resisting your tendency to strawman.
I’m not trying to strawman you, I’m suggesting a problem for how you stated you view so that you can reformulate your view in a way that I’ll better understand. I’m sorry if I wasn’t clear about that!
Newtonian physics is still true in its domain of applicability, it has never been true where it’s not been applicable, though people didn’t know this until 1905.
Right. But you said “‘accurate map’, a.k.a. ‘true map’ is a map that has been tested against the territory and found reliable enough to use as a guide for further travels”. My objection is that wide-applicability Newtonian Physics used to meet your criterion for truth (i.e., for a long time it passed all experimental tests and remained reliable for further research), but eventually stopped meeting it. Which suggests that it was true until it failed a test, or until it ceased to be a useful guide to further research; after that it became false. If you didn’t mean to suggest that, then I’m not sure I understand “map that has been tested against the territory and found reliable enough to use as a guide for further travels” anymore, which means I don’t know what you mean by “truth” and “accuracy” at this point.
Perhaps instead of defining “true” as “has been tested against the territory and found reliable enough to use as a guide for further travels”, what you meant to say was “has been tested against the territory and will always be found reliable enough to use as a guide for further travels”? That way various theories that had passed all tests at the time but are going to eventually fail them won’t count as ever having been ‘true’.
Numbers don’t physically exist because they are ideas, and as such belong in the realm of logic, not physics. (Again, I’m wearing a realist hat here.) I don’t think parsimony is required here. It’s a postulate, not a conclusion.
Postulates like ‘1 is nonphysical’, ‘2 is nonphysical’, etc. aren’t needed here; that would make our axiom set extraordinarily cluttered! The very idea that ‘ideas’ aren’t a part of the physical world is in no way obvious at the outset, much less axiomatic. There was a time when lightning seemed supernatural, a violation of the natural order; conceivably, we could have discovered that there isn’t really lightning (it’s some sort of illusion), but instead we discovered that it reduced to a physical process. Mental contents are like lightning. There may be another version of ‘idea’ or ‘thought’ or ‘abstraction’ that we can treat as a formalist symbol game or a useful fiction, but we still have to also either reduce or eliminate the natural-phenomenon-concept of abstract objects if we wish to advance the Great Reductionist Project.
It sounds like you want to eliminate them, and indeed stop even talking about them because they’re silly. I can get behind that, but only if we’re careful not to forget that not all mathematicians (etc.) agree on this point, and don’t equivocate between the two notions of ‘abstract’ (formal/fictive vs. spooky and metaphysical and Tegmarkish).
Then I don’t understand why you reply to questions of physical existence with some mathematical expressions...
Only because the apples are behaving like numbers whether you believe in numbers or not. You might not think our world does resemble the formalism in this respect, but that’s not obvious to everyone before we’ve talked the question over. A logic can be treated as a regimentation of natural language, or as an independent mathematical structure that happens to structurally resemble a lot of our informal reasoning and natural-language rules. Either way, information we get from logical analysis and deduction can tell us plenty about the physical world.
Re 2. First, let me adopt bounded realism here, with physics (external reality or territory) + logic (human models of reality, or maps). Let me ignore the ultraviolet divergence of decompartmentalization (hence “bounded”), where Many Words, Tegmark IV and modal realism are considered “territory”. To this end, let me put the UV cutoff on logic at the Popper’s boundary: only experimentally falsifiable maps are worth considering. A map is “true” means that it is an accurate representation of the piece of territory it is intended to represent. I apologize in advance if I am inventing new terms for the standard philosophical concepts—feel free to point me to the standard terminology.
I suspect you have, in fact, reinvented something. For reference, how does this “bounded realism” evaluate this statement:
On August 1st 2008 at midnight Greenwich time, a one-foot sphere of chocolate cake spontaneously formed in the center of the Sun; and then, in the natural course of events, this Boltzmann Cake almost instantly dissolved.
It makes no predictions; this is, in a sense, epiphenomenal cake—I know of no test we could perform that would distinguish between a world where this statement is false and one where it is true. Certainly tracking it provides us with no predictive power.
Yet is it somehow invalid? Is it gibberish? Can it be rejected a priori? Is there any sense in which it might be true? Is there any sense in which it might be false?
Sorry if I’m misinterpreting you here; I doubt this has much effect on your overall point.
How about this: Mathematicians have a conception of existence which is good enough for doing mathematics, but isn’t necessary correct. When you give a mathematical definition of existence, you are implicitly assuming a certain mathematical framework without justifying it. I think you would consider this criticism to be a variant of #2.
In particular, I also think about things mathematically, but when I do so, I don’t use first-order logic, but rather intuitionistic type theory. Can you give a definition for existence which would satisfy me?
I’m a mathematical fictionalist, so I’m happy to grant that there’s a good sense in which mathematical discourse isn’t strictly true, and doesn’t need to be.
Are you asking for a definition of an intuitionistic ‘exists’ predicate, or for the intuitionistic existential quantifier?
First, if you accept that mathematical constructs are fictional, why do you consider it valid to define a concept in terms of them? Second, I admit I wasn’t clear on this issue: The salient part of intuitionistic type theory isn’t intuitionism, but rather that it is a structural theory. This means that statements of the form “exists x, P(x)” are not well defined, but rather only statements of the form “exists x in A, P(x)” can be made.
I’m a little unclear on what your criticism is. Is one of these right?
You’re being too precise, whereas I wanted to have an informal discussion in terms of our everyday intuitions. So definitions are counterproductive; a little unclarity in what we mean is actually helpful for this topic.
There are two kinds of existence, one that holds for Plato’s Realm Of Invisible Mathy Things and one that holds for The Physical World. Your definitions may be true of the Mathy Things, but they aren’t true of things like apples and bumblebees. So you’re committing a category error.
I wanted you to give me a really rich, interesting explanation of what ‘existence’ is, in more fundamental terms. But instead you just copy-pasted a bland uninformative Standard Mathematical Logician Answer from some old textbook. That makes me sad. Please be more interesting next time.
If your point was 1, I’ll want to hear more. If it was 3, then my apologies! If it was 2, then I’ll have to disagree until I hear some argument as to why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)
Thank you, this framework helps. Definitely no to 1. Definitely yes to 2, with some corrections. Yes to some parts of 3.
Re 2. First, let me adopt bounded realism here, with physics (external reality or territory) + logic (human models of reality, or maps). Let me ignore the ultraviolet divergence of decompartmentalization (hence “bounded”), where Many Words, Tegmark IV and modal realism are considered “territory”. To this end, let me put the UV cutoff on logic at the Popper’s boundary: only experimentally falsifiable maps are worth considering. A map is “true” means that it is an accurate representation of the piece of territory it is intended to represent. I apologize in advance if I am inventing new terms for the standard philosophical concepts—feel free to point me to the standard terminology.
Again, “accurate map”, a.k.a. “true map” is a map that has been tested against the territory and found reliable enough to use as a guide for further travels, at least if one does not stray too far. Correspondingly, a piece of territory is said to “exist” if it is described by an accurate map.
On the other hand, your “invisible mathy things” live in the world of maps. Some of them use the same term “true”, but in a different way: given a set of rules of how to form strings of symbols, true statements are well-formed finite strings. They also use the same term “exist”, but also in a different way: given a set of rules, every well-formed string is said to “exist”.
Now, I am not a mathematician, so this may not be entirely accurate, but the gist is that conflating “exist” as applied to the territory and “exist” as applied to maps is indeed a category error. When someone talks about existence of physical objects and you write out something containing the existential quantifier, you are talking about a different category: not reality, but a subset of maps related to mathematical logic.
I am not sure whether this answers your objection that
but I hope it makes it clear why I find your replies unconvincing and generally not useful.
You’ve redefined ‘x exists’ to mean ‘x is described by a map that has been tested and so far has seemed reliable to us’, and ‘x is true’ correspondingly. One problem with this is that it’s historical: It commits us to saying ‘Newtonian physics used to be true, but these days it’s false (i.e., not completely reliable as a general theory)‘, and to saying ‘Phlogiston used to exist, but then it stopped existing because someone overturned phlogiston theory’. This is pretty strange.
Another problem is that it’s not clear what it takes to be ‘found reliable enough to use as a guide for further travels’. Surely there’s an important sense in which math is reliable in that sense, hence ‘true’ in the territory-ish sense you outlined above, not just in the map-ish sense. So perhaps we’ll need a more precise definition of territory-ish truth in order to clearly demonstrate why math isn’t in the territory, where the territory is defined by empirical adequacy.
I think your view, or one very close to yours, is actually a lot stronger (can be more easily defended, has broader implications) than your argument for it suggests. You can simply note that things like Abstract Numbers, being causally inert, couldn’t be responsible for the ‘unreasonable efficacy of mathematics’; so that efficacy can’t count as evidence for such Numbers. And nothing else is evidence for Numbers either. So we should conclude, on grounds of parsimony (perhaps fortified with anti-Tegmark’s-MUH arguments), that there are unlikely to be such Numbers. At that point, we can make the pragmatic, merely linguistic decision of saying that mathematicians are using ‘exists’ in a looser, more figurative sense.
Perhaps a few mathematicians are deluded into thinking that ‘exists’ means exactly the same thing in both contexts, but it is more charitable to interpret mathematics in general in the less ontologically committing way, because on the above arguments a platonistic mathematics would be little more than speculative theology. Basically, we end up with a formalist or fictionalist description of math, which I think is very plausible.
You see, we aren’t so different, you and I. Not once we bracket whether unexperienced cucumbers exist out there, anyway!
I disagree that this is a redefinition. You believe that elephants exists because you can go and see them, or talk to someone you trust who saw them, etc. You believe that live T-Rex (almost surely) does not exist because it went extinct some 60 odd million years ago. Both beliefs can be updated based on new information.
That’s not at all what I am saying. Consider resisting your tendency to strawman. Newtonian physics is still true in its domain of applicability, it has never been true where it’s not been applicable, though people didn’t know this until 1905.
Again, a belief at the time was that it existed, a more accurate belief (map) superseded the old one and now we know that phlogiston never existed. Maps thought of as being reliable can be found wanting all the time, so the territory they describe is no longer believed to exist, not stopped existing. This is pretty uncontroversial, I would think. Science didn’t kill gnomes and fairies, and such. At least this is the experiment-bounded realist position, as far as I understand it.
I can’t even parse that, sorry. Numbers don’t physically exist because they are ideas, and as such belong in the realm of logic, not physics. (Again, I’m wearing a realist hat here.) I don’t think parsimony is required here. It’s a postulate, not a conclusion.
Then I don’t understand why you reply to questions of physical existence with some mathematical expressions...
I’m not nearly as optimistic.
Sure, but ‘you believe in X because of Y’ does not as a rule let us conclude ‘X = Y’. I believe in elephants because of how they’ve causally impacted my experience, but I don’t believe that elephants are experiences of mine, or logical constructs out of my experiences and predictions. I believe elephants are animals.
Indeed, a large part of the reason I believe in elephants is that I think elephants would still exist even had you severed the causal links between me and them and I’d never learned about them. The territory doesn’t go away when you stop knowing about it, or even when you stop being able to ever know about it. If you shot an elephant in a rocket out of the observable universe, it wouldn’t stop existing, and I wouldn’t believe it had blinked out of existence or that questions regarding its existence were meaningless, once its future state ceased to be knowable to me.
Elephants don’t live in my map. But they also don’t live in my map-territory relation. Nor do they live in a function from observational data to hypotheses-that-help-us-build-rockets-and-iPhones-and-vaccines. They simply and purely live in the territory.
I’m not trying to strawman you, I’m suggesting a problem for how you stated you view so that you can reformulate your view in a way that I’ll better understand. I’m sorry if I wasn’t clear about that!
Right. But you said “‘accurate map’, a.k.a. ‘true map’ is a map that has been tested against the territory and found reliable enough to use as a guide for further travels”. My objection is that wide-applicability Newtonian Physics used to meet your criterion for truth (i.e., for a long time it passed all experimental tests and remained reliable for further research), but eventually stopped meeting it. Which suggests that it was true until it failed a test, or until it ceased to be a useful guide to further research; after that it became false. If you didn’t mean to suggest that, then I’m not sure I understand “map that has been tested against the territory and found reliable enough to use as a guide for further travels” anymore, which means I don’t know what you mean by “truth” and “accuracy” at this point.
Perhaps instead of defining “true” as “has been tested against the territory and found reliable enough to use as a guide for further travels”, what you meant to say was “has been tested against the territory and will always be found reliable enough to use as a guide for further travels”? That way various theories that had passed all tests at the time but are going to eventually fail them won’t count as ever having been ‘true’.
Postulates like ‘1 is nonphysical’, ‘2 is nonphysical’, etc. aren’t needed here; that would make our axiom set extraordinarily cluttered! The very idea that ‘ideas’ aren’t a part of the physical world is in no way obvious at the outset, much less axiomatic. There was a time when lightning seemed supernatural, a violation of the natural order; conceivably, we could have discovered that there isn’t really lightning (it’s some sort of illusion), but instead we discovered that it reduced to a physical process. Mental contents are like lightning. There may be another version of ‘idea’ or ‘thought’ or ‘abstraction’ that we can treat as a formalist symbol game or a useful fiction, but we still have to also either reduce or eliminate the natural-phenomenon-concept of abstract objects if we wish to advance the Great Reductionist Project.
It sounds like you want to eliminate them, and indeed stop even talking about them because they’re silly. I can get behind that, but only if we’re careful not to forget that not all mathematicians (etc.) agree on this point, and don’t equivocate between the two notions of ‘abstract’ (formal/fictive vs. spooky and metaphysical and Tegmarkish).
Only because the apples are behaving like numbers whether you believe in numbers or not. You might not think our world does resemble the formalism in this respect, but that’s not obvious to everyone before we’ve talked the question over. A logic can be treated as a regimentation of natural language, or as an independent mathematical structure that happens to structurally resemble a lot of our informal reasoning and natural-language rules. Either way, information we get from logical analysis and deduction can tell us plenty about the physical world.
I suspect you have, in fact, reinvented something. For reference, how does this “bounded realism” evaluate this statement:
It makes no predictions; this is, in a sense, epiphenomenal cake—I know of no test we could perform that would distinguish between a world where this statement is false and one where it is true. Certainly tracking it provides us with no predictive power.
Yet is it somehow invalid? Is it gibberish? Can it be rejected a priori? Is there any sense in which it might be true? Is there any sense in which it might be false?
Sorry if I’m misinterpreting you here; I doubt this has much effect on your overall point.
How about this: Mathematicians have a conception of existence which is good enough for doing mathematics, but isn’t necessary correct. When you give a mathematical definition of existence, you are implicitly assuming a certain mathematical framework without justifying it. I think you would consider this criticism to be a variant of #2.
In particular, I also think about things mathematically, but when I do so, I don’t use first-order logic, but rather intuitionistic type theory. Can you give a definition for existence which would satisfy me?
I’m a mathematical fictionalist, so I’m happy to grant that there’s a good sense in which mathematical discourse isn’t strictly true, and doesn’t need to be.
Are you asking for a definition of an intuitionistic ‘exists’ predicate, or for the intuitionistic existential quantifier?
(Note: I added a link in my previous comment)
First, if you accept that mathematical constructs are fictional, why do you consider it valid to define a concept in terms of them? Second, I admit I wasn’t clear on this issue: The salient part of intuitionistic type theory isn’t intuitionism, but rather that it is a structural theory. This means that statements of the form “exists x, P(x)” are not well defined, but rather only statements of the form “exists x in A, P(x)” can be made.