Well, leaving out mathematical principles, as explained in the text, means that the universe can be defined in a consistent manner. That’s really useful for cosmologists, astrophysicists, and other people such as Prof Tegmark, trying to understand what universes could exist, if other universes existed before our own and generally do experiments to understand the scope of the universe. Maybe it’s not everyone’s trade, but for many people it really matters.
Putting it another way, adding mathematical concepts in the way described, without limiting the scope or doing something to handle the Cantor paradox, leads to a definition of the universe that’s logically inconsistent (ie it always leaves out something, so its a proof by contradiction, simply, the universe can’t be ‘large enough’ to encapsulate those entities.
Am I saying the universe exemplifies the maximum set of consistent maths? Not at all! Quite the opposite in fact—I’m not sure how you got that from this summary, as I’m saying the opposite, that many mathematical objects to not end up being exemplified, or related. The cardinality is the same but the set of things not mapping to anything in the universe is much much larger.
Am I saying the mathematical universe is inconsistent if it exists at all? Well, not really—I don’t know if there is a ‘mathematical universe’ per se. All we know is it’s too big to fit in our universe and stuff gets left out. We know that its not as large as ‘anything I make up’ because there are theorems that lead to contradictions and that limits the mathematical objects that are ‘real’ (in the mathematical sense). I imagine that it’s more like a ‘wood between the worlds’ or like a kind of meta language that is kind of between universes (if there are other universes) or otherwise simply exists outside (if there’s only one universe). So I don’t see this as an argument against realism—as realism was used to posit the notion of existence to ‘mathematically consistent’ in the first place, so it couldn’t go in that direction.
Totally agree that mathematics being a kind of encapsulated universe consistent with itself would undermine its own truth, and thereby be automatically false because one is relying on logic in the first place to posit it. So, indeed, that’s not a stance to take here. In fact, it’s a good argument for absolute truth, and one I’m fond of using.
‘Smaller but consistent’ MU—I’m not sure what you mean here—the ‘smallness’ would come from leaving out non-truths, ie contradictions, and that would be needed ti preserve a sense of truth in any framework. But also I think MU (does that mean ‘mathematical universe’) is a potentially misleading term as I just described above.
The overall result of this work can thus be summarised by the statement that ‘abstract objects can be demonstrated to be in a distinct universe from ours, as the set ofboth of them together cannot be defined in a consistent way logically.’
That is necessary, but insufficient , at best. If mathematical entities don’t have real existence, as fictionalists cliaim, then there is also no real inconsistency.
Comment: “That is necessary, but insufficient , at best. If mathematical entities don’t have real existence, as fictionalists cliaim, then there is also no real inconsistency.”
Response: Can you say that more precisely, what is necessary but insufficient, for what?
What I stated was, from this work, based on the assumptions as laid out in the manuscript, I can verbally summarise the result by the statement that ‘abstract objects can be demonstrated to be in a distinct universe from ours, as the set of both of them together cannot be defined in a consistent way logically.’
Do you mean that these assumptions, together with the steps, don’t show that these abstract objects to be in a universe distinct from ours, in the context of this formalism, and the definitions that I have supplied? OR do you mean that this statement, is necessary but insufficient to show a separate thing, ie to prove or to demonstrate some of the other concepts we have been discussing, such as MR, which I stated I did not try to prove? ie as an encapsulated logic I don’t see where I went wrong.
Based on your statement, “If mathematical entities don’t have real existence, as fictionalists cliaim, then there is also no real inconsistency.”, I don’t need to rely on there being a ‘real’ inconsistency for the steps to still work. What I mean is, what is ‘real’ or not is just the background metaphysic. A mathematical fictionalist would still hold that there are proofs and theorems, just that they don’t have a ‘real’ existence. But, I haven’t taken the assumption of mathematical fictionalism, in which I don’t know if I can talk about abstract objects at all—so it doesn’t apply here.
When I discuss mathematical fictionalism (note, this is not something I talk about in the manuscript), I have some questions I need to clarify about how it works, and seek guidance on, to be able to argue effectively on that topic as I haven’t had it fully described to me. As I keep mentioning, where I am at, provisionally, at the moment, is that I keep running into that said Solipsism (please note, as mentioned, this is not the argument from my paper which assumes MR). So, can we resolve it—in the primordial universe, how are we to understand the subatomic particles being arranged into groups, and so forth, if these structures required for our existence only are fictions occurring within the mind of people? (I assume that’s what fictionalism means)… if you can clarify
Can you say that more precisely, what is necessary but insufficient, for what?
The assumption, alone, that the physical universe must be consistent is insufficient to show that there is a separate MU.
The assumption that the physical universe must be consistent, plus the assumption of MR, plus the assumption that consistency doesn’t matter for MU’s is sufficient to show that there is a separate MU.
‘abstract objects can be demonstrated to be in a distinct universe from ours, as the set of both of them together cannot be defined in a consistent way logically.’
But you have only shown that that follows from your assumptions, not that it is actually true, ie. your assumptions are actually true.
Do you mean that these assumptions, together with the steps, don’t show that these abstract objects to be in a universe distinct from ours, in the context of this formalism, and the definitions that I have supplied?
No, it’s valid, but it’s not sound. Valid but unsound arguments are ten a penny—you can show almost anything given three arbitrary premises.
But, I haven’t taken the assumption of mathematical fictionalism, in which I don’t know if I can talk about abstract objects at all—so it doesn’t apply here.
Fictionalism isn’t known to be true or false, neither is realism. It’s an open subject. You can make arguments against fictionalism, the fictionalist can make arguments against realism.
You seem to be hinting at an argument where an entity has to really exist in order to be talked about (an argument you haven’t brought before). But the plain fact is that we can talk about fictional entities, like Gandalf or Sherlock Holmes—so the fictionalist has a robust response.
I have some questions I need to clarify about how it works, and seek guidance on, to be able to argue effectively on that topic as I haven’t had it fully described to me.
Can you not learn that from books? How did you learn about realism?
As I keep mentioning, where I am at, provisionally, at the moment, is that I keep running into that said Solipsism
Even now? But you’ve agreed that the existence of mathematically describable entities outside the mind doesn’t imply the existence of ontologically mathematical entities outside the mind .
So, can we resolve it—in the primordial universe, how are we to understand the subatomic particles being arranged into groups, and so forth, if these structures required for our existence only are fictions occurring within the mind of people? (I assume that’s what fictionalism means)… if you can clarify
Imagine the library of Babel. Since it contains every story, some stories will be history, whilst most will be fantasy. But the true stories are still stories in books.
Hi Tag, I didn’t hear back from you so I suppose I can assume that my explanations were satisfactory and I had now resolved the qualms as previously raised. Thank you for your attention.
Comment: 1: “The assumption, alone, that the physical universe must be consistent is insufficient to show that there is a separate MU.
The assumption that the physical universe must be consistent, plus the assumption of MR, plus the assumption that consistency doesn’t matter for MU’s is sufficient to show that there is a separate MU.”
Response 1: Thanks for clarifying what you mean here. I hear you, but I have been clear in the manuscript that I take the assumption of MR. And then, for assumption of the physical universe needing to be consistent isn’t really assumed, but it’s discussed, and I bring to bear a practicality argument—that it’s actually a semantic point (defining the scope of the term ‘universe’) and so I am clear here also. And the third item you mention, that it is required to assume that consistency doesn’t matter for MU: I’m not sure I agree here. I don’t think I made statements either way as to the consistency of a MU (or multiple MUs) in the work. We did discuss that consistency can be considered dependent on whether one holds one maximal MU or multiple separate MU, but I don’t think an assumption on that is needed with regards to the scope of the argument pertaining to objects being outside the scope of the (physical) universe.
Comment 2: “But you have only shown that that follows from your assumptions, not that it is actually true, ie. your assumptions are actually true.”
Response 2: Well, that’s something still. What would be required as an additional piece you would like to have available is a proof that MR is true. I’m not sure if such a thing exists, as it might be axiomatic. It’s definitely not in the scope I laid down in the manuscript but could be an interesting new topic. Hm, I wonder if it ends up being that multiple ontological views are proved valid, but it ends up being more convenient (ie less ‘epicycles’) to adopt one of them in particular and so from an Occam’s Razor perspective, one comes to the fore. But that’s speculation. Interesting point though!
Response 3: Ah, but that’s all I an supplying. I am sorry the work was not all you would hope it to be, right now. But these things take time and effort and that’s as far as we have got so far, unless you wish to collaborate on an extension as described above—though, I’m not sure there is an easy resolution especially if there’s a semantic component described above.
Context: But, I haven’t taken the assumption of mathematical fictionalism, in which I don’t know if I can talk about abstract objects at all—so it doesn’t apply here.
Comment 4: “Fictionalism isn’t known to be true or false, neither is realism. It’s an open subject. You can make arguments against fictionalism, the fictionalist can make arguments against realism.
You seem to be hinting at an argument where an entity has to really exist in order to be talked about (an argument you haven’t brought before). But the plain fact is that we can talk about fictional entities, like Gandalf or Sherlock Holmes—so the fictionalist has a robust response.”
Response 4: Ah you have misunderstood me, what I meant was I don’t have a detailed knowledge of how to construct a framework in mathematical fictionalism, and if I can define an object in that framework and refer to it in quite the same way. I don’t mean that a Sherlock Holmes or Gandalf can’t be referred to in a fictionalism, it’s just that I don’t know enough about how it set up (what os the nature of the ‘fiction’ from an ontological perspective, to know what properties it should have, or if the target of the reference is the sort of thing that can have properties (I’m assuming it can?). In MR, it seems more straightforward that if I can define an entity, it can be given an existence in an abstract sense, for the object referred to to have properties (including non-extantness) but without requiring that the object have a related extant object in the universe (such as a Sherlock Holmes who is clearly not extant even though his concept exists).
Context: I have some questions I need to clarify about how it works, and seek guidance on, to be able to argue effectively on that topic as I haven’t had it fully described to me.
Comment 5: “Can you not learn that from books? How did you learn about realism?”
Response 5: Well yes, I can, but it would be a new research topic and these things take time, as mentioned. One has to start somewhere and one can’t expect all parts of a project to be done right away, especially as research topics need to unfold incrementally to build a body of knowledge. But what you say here goes beyond just reading—I don’t know the body of literature of fictionalism/realism contains a systemisation in the way I have described in order to compare the variants that could be done. I think I would have to do that work, and so I have started here: but I picked realism because it’s a more natural starting point as mentioned, from a physics perspective and a view (at least tacitly) held by colleagues and even if not admitted, is operated under when doing daily life in the world of physics research.
Context: As I keep mentioning, where I am at, provisionally, at the moment, is that I keep running into that said Solipsism
Comment 6: “Even now? But you’ve agreed that the existence of mathematically describable entities outside the mind doesn’t imply the existence of ontologically mathematical entities outside the mind .”
Response 6: I mean, I haven’t investigated down to the very bottom what precisely a ‘mathematically describable entity outside the mind but not leading to an ontological mathematical entity’ really is, and if it doesn’t lead to other issues, contradictions, etc down the track. I can see that it’s something that can be proposed but I don’t know if such a thing is sound, hence why the investigation as I mentioned is taking place. It sounds like it is resolved in your mind but I haven’t understood it or had it presented to me yet. It’s not enough to state a thing could be so, we need to check the consistency through and through and I haven’t seen a detailed presentation of such a thing yet (if it exists).
Context: So, can we resolve it—in the primordial universe, how are we to understand the subatomic particles being arranged into groups, and so forth, if these structures required for our existence only are fictions occurring within the mind of people? (I assume that’s what fictionalism means)… if you can clarify
Comment 7: “Imagine the library of Babel. Since it contains every story, some stories will be history, whilst most will be fantasy. But the true stories are still stories in books.”
Response 7: Thank you for this metaphor, it is helpful—here in the library of Babel, which contains every story, only some of the stories will be history, and some are various other non-historical genres. This is similar to the statement in the work I was doing, where ‘extantness’ means ‘history’ ie the abstraction projects down to relate to an actual object in the real world. Whereas there might be many abstractions (existing but not ‘extant’ and also non-existing entities means that defining them leads to a contradiction and can’t be defined consistently). So indeed only a small portion of the abstractions have the ‘extant’ property (using the world extant to distinguish it from ‘exist’ in the mathematical sense), and what I did in my formalism was to create a meta-language scoped so that the objects only ‘exist’ if they are ‘extant’, which places some limits and also structure and properties for the formalism.
But anyway, the true stories (the ‘histories’) or the ‘extant’ items, are still abstractions we refer to, it’s just that they have this special projective relationship with a physical object. The fact that the formalism can only ever speak in abstractions is part and parcel of having a self-contained meta-language (and also useful as we can do inquiries on it), but the fact that you always need a ‘pointer’ to be able to included in the formalism was precisely why the Labeling Principle was imposed, which defines this property.
So, here in your metaphor, it seems very consistent with the view I have, and relied upon in the work, so I don’t quite follow why the metaphor is against the view expounded somehow. I agree with the metaphor, and it is the same as my view.
Well, leaving out mathematical principles, as explained in the text, means that the universe can be defined in a consistent manner.
The universe meaning the physical universe? But that’s an argument against MR.
Putting it another way, adding mathematical concepts in the way described, without limiting the scope or doing something to handle the Cantor paradox, leads to a definition of the universe that’s logically inconsistent
Which universe? Are you talking about a single maths+physics universe? (as opposed to Platonism)
Am I saying the mathematical universe is inconsistent if it exists at all? Well, not really—I don’t know if there is a ‘mathematical universe’ per se.
You can specify which kind of MU you are talking about, as a hypothesis. (although at this point I am not even clear whether you are for or against MR.
Totally agree that mathematics being a kind of encapsulated universe consistent with itself would undermine its own truth,
I didn’t say that. The problem is that it doesn’t make any difference to our activities—we can’t tell whether the axiom of choice is true by peaking into Plato’s heaven. So it’s a huge amount of additional entities that don’t do anything, in practice.
We know that its not as large as ‘anything I make up’ because there are theorems that lead to contradictions and that limits the mathematical objects that are ‘real’ (in the mathematical sense).
...and there isn’t a unique non contradictory MU either.
So I don’t see this as an argument against realism
The argument is that the MU is either maximal and inconsistent, or non-unique., ie indefineable. Both are problems.
Plus the Occam’s razor problem.
‘Smaller but consistent’ MU—I’m not sure what you mean here—the ‘smallness’ would come from leaving out non-truths, ie contradictions,
It would come from leaving out one side of a contrdiction, ie. there are two smaller universes for every contradiction.
But also I think MU (does that mean ‘mathematical universe’) is a potentially misleading term as I just described above.
Responses to questions, also thank you for your patience—there seems to be a misunderstanding under the surface that we are both approaching, perhaps in terms of terminology that is causing some of the disagreement:
Comment 1: “The universe meaning the physical universe? But that’s an argument against MR.”
Response 1: Yes, the universe meaning the physical universe, and I am taking Mathematical Realism to mean that the mathematical statements are real (they exist) but not in the physical universe. If one defined Mathematical Realism as compelling the inclusion of mathematical objects into the physical universe, then I don’t hold that. This position is about demonstrating how the putting together of mathematical objects in the physical universe means the new composite universe can’t be consistently defined. So the real mathematical objects have to be outside the universe.
Context: Putting it another way, adding mathematical concepts in the way described, without limiting the scope or doing something to handle the Cantor paradox, leads to a definition of the universe that’s logically inconsistent
Comment 2: “Which universe? Are you talking about a single maths+physics universe? (as opposed to Platonism)”
Response 2: Leading to a definition of a universe that is a single maths+physics universe, which can’t be defined consistency. Which means we end up dispensing with it and then having something closer to Platonism.
Context: Am I saying the mathematical universe is inconsistent if it exists at all? Well, not really—I don’t know if there is a ‘mathematical universe’ per se.
Comment 3: “You can specify which kind of MU you are talking about, as a hypothesis. (although at this point I am not even clear whether you are for or against MR.”
Response 3: I don’t specify a MU as I don’t assume such a one exists, in an encapsulated way. I specify that we hold that mathematical truths are real (exist) and show that they need to lie outside the physical universe. I think this must be the heart of the confusion—there must be a definition of MR that is different from the way I am using it. It appears as though, in your view, that having something ‘exist’ means it needs to be encapsulated either in a) a universe, or b) the physical universe so they ar combined together. (Not sure which you are assuming), whereas I had not taken MR to mean this, otherwise the work wouldn’t make sense. Apart from the terminology, I rely on the definition of the physical universe in the manuscript, ie Eq(68).
Context: Totally agree that mathematics being a kind of encapsulated universe consistent with itself would undermine its own truth,
Comment 4: “I didn’t say that. The problem is that it doesn’t make any difference to our activities—we can’t tell whether the axiom of choice is true by peaking into Plato’s heaven. So it’s a huge amount of additional entities that don’t do anything, in practice.”
Response 4: Ok, we can’t tell if an axiom is true (by definition of what an axiom is) as its one of the building blocks for a particular computational logic we may choose to define, L. Now, if L is embedded in some other framework, a meta language, we may be able to make some statements about the axioms, but not from within L itself. But, for a given L, and its axioms, the truths would be statements that are formatted as theorems, which can either be proved or take the format of a Goedel sentence (ie only proved from outside L but not from within, unless the consistency of L is undermined). In other words, the mathematical truths and realities that we can ‘peek into Plato’s heaven’ to see, aren’t the axioms, of a given L (in the context of L, unless we go outside of L), they are of the format “given a and b” (where we can make ‘a’ and ‘b’ anything we want), “then c follows”. So I am stating that such a truth holds true, for my selection of ‘a’ and ‘b’, regardless of whether one is in a physical universe or any other speculated other physical universe. ie I am asserting that these mathematical truths hold in any physical universe as a meta language, as I can choose anything I want for ‘a’ and ‘b’, then logic of ‘c’ holding up in the context of the system ‘L’ I have defined, is immutable and true.
Context: We know that its not as large as ‘anything I make up’ because there are theorems that lead to contradictions and that limits the mathematical objects that are ‘real’ (in the mathematical sense).
Comment 5 ”...and there isn’t a unique non contradictory MU either.”
Response 5: Fair—indeed there wouldn’t be, as there are many different systems of logic, say, I could define (different L’s, as mentioned above, for instance) that could all be internally consistent. I don’t however state that there is such an MU encapsulated. I only really explore the physical universe and whether we can bolt-on mathematical truths to it or not. Believing mathematical theorems are real truths that hold everywhere (regardless of what universe), and finding that we make the definition of the universe tricky if we definite as the physical universe + mathematical truths, I concluded that, adding mathematical all truths enters a Cantor universality paradox. However, you could for example, argue for a subset of mathematical truths (just the ones that apply to the physical universe in some way) and bolt it on to the physical universe to make a new type of MR. I haven’t seen a prescription for how this could be done though in practice. e.g. it’s not clear a given phenomenon can even have a single mathematical prescription applied to it. I could use complex numbers, or matrices, or other types of vector spaces, to achieve a similar end—the mathematical machinery isn’t necessarily unique when applied to their relevant targets in the physical universe, and yet all these mathematical theorems and the objects they operate on are different entities—so where does it end? (ie how can we prescribe how to limit the mathematical theorems that might get bolted on to the physical universe to make a new universe including them all, so as not to run into a Cantor paradox)?
Context: So I don’t see this as an argument against realism
Comment 6: “The argument is that the MU is either maximal and inconsistent, or non-unique., ie indefineable. Both are problems.
Plus the Occam’s razor problem.”
Response 6: If i understand correctly what MU means here, that is some set that includes at least enough mathematical objects to encounter a Cantor paradox), then I agree an MU being maximal makes it undefinable consistently. But the other case that, if it is split into smaller/limited subsets that don’t encounter that, then there are a multiplicity of them, and not a unique MU, I don’t know if this second option is a separate problem though. If I arbitrarily separate them out into different components, and I select one of them (perhaps a candidate to bolt onto the physical universe to define a phys+math universe), I can still ask if the remaining ones I left out are ‘true’ or ‘real’ and whether they are in this newly defined universe.
I don’t know of any prescription for selecting such a limited candidate subset though.
Context: ‘Smaller but consistent’ MU—I’m not sure what you mean here—the ‘smallness’ would come from leaving out non-truths, ie contradictions,
Comment 7: “It would come from leaving out one side of a contrdiction, ie. there are two smaller universes for every contradiction.”
Response 7: Ok, which amounts to defining a smaller calculus, like an ‘L’, which has only certain axioms, contains only certain structures and is limited to only certain theorems.
Context: But also I think MU (does that mean ‘mathematical universe’) is a potentially misleading term as I just described above.
Comment 8: “So.,,,have you a better one?”
Response 8: I was actually going to ask you the same question, but for MR—if it appears my definition of MR is different from what you had in mind, what should I call my thing, if the naming convention is causing confusion? Essentially what I meant was the math I wrote—the words are hard to get right as people come at the problem with different background of terminology usage.
For your question here about a better term for MU, I think we can retain it but clarify the different concepts. In one case, MU could be used to mean an encapsulated system (ie a universe) of mathematical theorems. Or in one case, it could mean one of the completely consistent definable subsets of it. In another case, it could mean the physical universe + a bolt on of either all or a consistently definable subset of mathematical theorems (ie both physical + math together). Maybe we should call the latter, ‘PMU’?
Well, leaving out mathematical principles, as explained in the text, means that the universe can be defined in a consistent manner. That’s really useful for cosmologists, astrophysicists, and other people such as Prof Tegmark, trying to understand what universes could exist, if other universes existed before our own and generally do experiments to understand the scope of the universe. Maybe it’s not everyone’s trade, but for many people it really matters.
Putting it another way, adding mathematical concepts in the way described, without limiting the scope or doing something to handle the Cantor paradox, leads to a definition of the universe that’s logically inconsistent (ie it always leaves out something, so its a proof by contradiction, simply, the universe can’t be ‘large enough’ to encapsulate those entities.
Am I saying the universe exemplifies the maximum set of consistent maths? Not at all! Quite the opposite in fact—I’m not sure how you got that from this summary, as I’m saying the opposite, that many mathematical objects to not end up being exemplified, or related. The cardinality is the same but the set of things not mapping to anything in the universe is much much larger.
Am I saying the mathematical universe is inconsistent if it exists at all? Well, not really—I don’t know if there is a ‘mathematical universe’ per se. All we know is it’s too big to fit in our universe and stuff gets left out. We know that its not as large as ‘anything I make up’ because there are theorems that lead to contradictions and that limits the mathematical objects that are ‘real’ (in the mathematical sense). I imagine that it’s more like a ‘wood between the worlds’ or like a kind of meta language that is kind of between universes (if there are other universes) or otherwise simply exists outside (if there’s only one universe). So I don’t see this as an argument against realism—as realism was used to posit the notion of existence to ‘mathematically consistent’ in the first place, so it couldn’t go in that direction.
Totally agree that mathematics being a kind of encapsulated universe consistent with itself would undermine its own truth, and thereby be automatically false because one is relying on logic in the first place to posit it. So, indeed, that’s not a stance to take here. In fact, it’s a good argument for absolute truth, and one I’m fond of using.
‘Smaller but consistent’ MU—I’m not sure what you mean here—the ‘smallness’ would come from leaving out non-truths, ie contradictions, and that would be needed ti preserve a sense of truth in any framework. But also I think MU (does that mean ‘mathematical universe’) is a potentially misleading term as I just described above.
That is necessary, but insufficient , at best. If mathematical entities don’t have real existence, as fictionalists cliaim, then there is also no real inconsistency.
Comment: “That is necessary, but insufficient , at best. If mathematical entities don’t have real existence, as fictionalists cliaim, then there is also no real inconsistency.”
Response: Can you say that more precisely, what is necessary but insufficient, for what?
What I stated was, from this work, based on the assumptions as laid out in the manuscript, I can verbally summarise the result by the statement that ‘abstract objects can be demonstrated to be in a distinct universe from ours, as the set of both of them together cannot be defined in a consistent way logically.’
Do you mean that these assumptions, together with the steps, don’t show that these abstract objects to be in a universe distinct from ours, in the context of this formalism, and the definitions that I have supplied? OR do you mean that this statement, is necessary but insufficient to show a separate thing, ie to prove or to demonstrate some of the other concepts we have been discussing, such as MR, which I stated I did not try to prove? ie as an encapsulated logic I don’t see where I went wrong.
Based on your statement, “If mathematical entities don’t have real existence, as fictionalists cliaim, then there is also no real inconsistency.”, I don’t need to rely on there being a ‘real’ inconsistency for the steps to still work. What I mean is, what is ‘real’ or not is just the background metaphysic. A mathematical fictionalist would still hold that there are proofs and theorems, just that they don’t have a ‘real’ existence. But, I haven’t taken the assumption of mathematical fictionalism, in which I don’t know if I can talk about abstract objects at all—so it doesn’t apply here.
When I discuss mathematical fictionalism (note, this is not something I talk about in the manuscript), I have some questions I need to clarify about how it works, and seek guidance on, to be able to argue effectively on that topic as I haven’t had it fully described to me. As I keep mentioning, where I am at, provisionally, at the moment, is that I keep running into that said Solipsism (please note, as mentioned, this is not the argument from my paper which assumes MR). So, can we resolve it—in the primordial universe, how are we to understand the subatomic particles being arranged into groups, and so forth, if these structures required for our existence only are fictions occurring within the mind of people? (I assume that’s what fictionalism means)… if you can clarify
The assumption, alone, that the physical universe must be consistent is insufficient to show that there is a separate MU.
The assumption that the physical universe must be consistent, plus the assumption of MR, plus the assumption that consistency doesn’t matter for MU’s is sufficient to show that there is a separate MU.
But you have only shown that that follows from your assumptions, not that it is actually true, ie. your assumptions are actually true.
No, it’s valid, but it’s not sound. Valid but unsound arguments are ten a penny—you can show almost anything given three arbitrary premises.
https://en.wikipedia.org/wiki/Validity_(logic)
Fictionalism isn’t known to be true or false, neither is realism. It’s an open subject. You can make arguments against fictionalism, the fictionalist can make arguments against realism.
You seem to be hinting at an argument where an entity has to really exist in order to be talked about (an argument you haven’t brought before). But the plain fact is that we can talk about fictional entities, like Gandalf or Sherlock Holmes—so the fictionalist has a robust response.
Can you not learn that from books? How did you learn about realism?
Even now? But you’ve agreed that the existence of mathematically describable entities outside the mind doesn’t imply the existence of ontologically mathematical entities outside the mind .
Imagine the library of Babel. Since it contains every story, some stories will be history, whilst most will be fantasy. But the true stories are still stories in books.
Hi Tag, I didn’t hear back from you so I suppose I can assume that my explanations were satisfactory and I had now resolved the qualms as previously raised. Thank you for your attention.
Comment: 1: “The assumption, alone, that the physical universe must be consistent is insufficient to show that there is a separate MU.
The assumption that the physical universe must be consistent, plus the assumption of MR, plus the assumption that consistency doesn’t matter for MU’s is sufficient to show that there is a separate MU.”
Response 1: Thanks for clarifying what you mean here. I hear you, but I have been clear in the manuscript that I take the assumption of MR. And then, for assumption of the physical universe needing to be consistent isn’t really assumed, but it’s discussed, and I bring to bear a practicality argument—that it’s actually a semantic point (defining the scope of the term ‘universe’) and so I am clear here also. And the third item you mention, that it is required to assume that consistency doesn’t matter for MU: I’m not sure I agree here. I don’t think I made statements either way as to the consistency of a MU (or multiple MUs) in the work. We did discuss that consistency can be considered dependent on whether one holds one maximal MU or multiple separate MU, but I don’t think an assumption on that is needed with regards to the scope of the argument pertaining to objects being outside the scope of the (physical) universe.
Comment 2: “But you have only shown that that follows from your assumptions, not that it is actually true, ie. your assumptions are actually true.”
Response 2: Well, that’s something still. What would be required as an additional piece you would like to have available is a proof that MR is true. I’m not sure if such a thing exists, as it might be axiomatic. It’s definitely not in the scope I laid down in the manuscript but could be an interesting new topic. Hm, I wonder if it ends up being that multiple ontological views are proved valid, but it ends up being more convenient (ie less ‘epicycles’) to adopt one of them in particular and so from an Occam’s Razor perspective, one comes to the fore. But that’s speculation. Interesting point though!
Comment 3: “No, it’s valid, but it’s not sound. Valid but unsound arguments are ten a penny—you can show almost anything given three arbitrary premises. https://en.wikipedia.org/wiki/Validity_(logic)”
Response 3: Ah, but that’s all I an supplying. I am sorry the work was not all you would hope it to be, right now. But these things take time and effort and that’s as far as we have got so far, unless you wish to collaborate on an extension as described above—though, I’m not sure there is an easy resolution especially if there’s a semantic component described above.
Context: But, I haven’t taken the assumption of mathematical fictionalism, in which I don’t know if I can talk about abstract objects at all—so it doesn’t apply here.
Comment 4: “Fictionalism isn’t known to be true or false, neither is realism. It’s an open subject. You can make arguments against fictionalism, the fictionalist can make arguments against realism.
You seem to be hinting at an argument where an entity has to really exist in order to be talked about (an argument you haven’t brought before). But the plain fact is that we can talk about fictional entities, like Gandalf or Sherlock Holmes—so the fictionalist has a robust response.”
Response 4: Ah you have misunderstood me, what I meant was I don’t have a detailed knowledge of how to construct a framework in mathematical fictionalism, and if I can define an object in that framework and refer to it in quite the same way. I don’t mean that a Sherlock Holmes or Gandalf can’t be referred to in a fictionalism, it’s just that I don’t know enough about how it set up (what os the nature of the ‘fiction’ from an ontological perspective, to know what properties it should have, or if the target of the reference is the sort of thing that can have properties (I’m assuming it can?). In MR, it seems more straightforward that if I can define an entity, it can be given an existence in an abstract sense, for the object referred to to have properties (including non-extantness) but without requiring that the object have a related extant object in the universe (such as a Sherlock Holmes who is clearly not extant even though his concept exists).
Context: I have some questions I need to clarify about how it works, and seek guidance on, to be able to argue effectively on that topic as I haven’t had it fully described to me.
Comment 5: “Can you not learn that from books? How did you learn about realism?”
Response 5: Well yes, I can, but it would be a new research topic and these things take time, as mentioned. One has to start somewhere and one can’t expect all parts of a project to be done right away, especially as research topics need to unfold incrementally to build a body of knowledge. But what you say here goes beyond just reading—I don’t know the body of literature of fictionalism/realism contains a systemisation in the way I have described in order to compare the variants that could be done. I think I would have to do that work, and so I have started here: but I picked realism because it’s a more natural starting point as mentioned, from a physics perspective and a view (at least tacitly) held by colleagues and even if not admitted, is operated under when doing daily life in the world of physics research.
Context: As I keep mentioning, where I am at, provisionally, at the moment, is that I keep running into that said Solipsism
Comment 6: “Even now? But you’ve agreed that the existence of mathematically describable entities outside the mind doesn’t imply the existence of ontologically mathematical entities outside the mind .”
Response 6: I mean, I haven’t investigated down to the very bottom what precisely a ‘mathematically describable entity outside the mind but not leading to an ontological mathematical entity’ really is, and if it doesn’t lead to other issues, contradictions, etc down the track. I can see that it’s something that can be proposed but I don’t know if such a thing is sound, hence why the investigation as I mentioned is taking place. It sounds like it is resolved in your mind but I haven’t understood it or had it presented to me yet. It’s not enough to state a thing could be so, we need to check the consistency through and through and I haven’t seen a detailed presentation of such a thing yet (if it exists).
Context: So, can we resolve it—in the primordial universe, how are we to understand the subatomic particles being arranged into groups, and so forth, if these structures required for our existence only are fictions occurring within the mind of people? (I assume that’s what fictionalism means)… if you can clarify
Comment 7: “Imagine the library of Babel. Since it contains every story, some stories will be history, whilst most will be fantasy. But the true stories are still stories in books.”
Response 7: Thank you for this metaphor, it is helpful—here in the library of Babel, which contains every story, only some of the stories will be history, and some are various other non-historical genres. This is similar to the statement in the work I was doing, where ‘extantness’ means ‘history’ ie the abstraction projects down to relate to an actual object in the real world. Whereas there might be many abstractions (existing but not ‘extant’ and also non-existing entities means that defining them leads to a contradiction and can’t be defined consistently). So indeed only a small portion of the abstractions have the ‘extant’ property (using the world extant to distinguish it from ‘exist’ in the mathematical sense), and what I did in my formalism was to create a meta-language scoped so that the objects only ‘exist’ if they are ‘extant’, which places some limits and also structure and properties for the formalism. But anyway, the true stories (the ‘histories’) or the ‘extant’ items, are still abstractions we refer to, it’s just that they have this special projective relationship with a physical object. The fact that the formalism can only ever speak in abstractions is part and parcel of having a self-contained meta-language (and also useful as we can do inquiries on it), but the fact that you always need a ‘pointer’ to be able to included in the formalism was precisely why the Labeling Principle was imposed, which defines this property. So, here in your metaphor, it seems very consistent with the view I have, and relied upon in the work, so I don’t quite follow why the metaphor is against the view expounded somehow. I agree with the metaphor, and it is the same as my view.
The universe meaning the physical universe? But that’s an argument against MR.
Which universe? Are you talking about a single maths+physics universe? (as opposed to Platonism)
You can specify which kind of MU you are talking about, as a hypothesis. (although at this point I am not even clear whether you are for or against MR.
I didn’t say that. The problem is that it doesn’t make any difference to our activities—we can’t tell whether the axiom of choice is true by peaking into Plato’s heaven. So it’s a huge amount of additional entities that don’t do anything, in practice.
...and there isn’t a unique non contradictory MU either.
The argument is that the MU is either maximal and inconsistent, or non-unique., ie indefineable. Both are problems.
Plus the Occam’s razor problem.
It would come from leaving out one side of a contrdiction, ie. there are two smaller universes for every contradiction.
So.,,,have you a better one?
Responses to questions, also thank you for your patience—there seems to be a misunderstanding under the surface that we are both approaching, perhaps in terms of terminology that is causing some of the disagreement:
Comment 1: “The universe meaning the physical universe? But that’s an argument against MR.”
Response 1: Yes, the universe meaning the physical universe, and I am taking Mathematical Realism to mean that the mathematical statements are real (they exist) but not in the physical universe. If one defined Mathematical Realism as compelling the inclusion of mathematical objects into the physical universe, then I don’t hold that. This position is about demonstrating how the putting together of mathematical objects in the physical universe means the new composite universe can’t be consistently defined. So the real mathematical objects have to be outside the universe.
Comment 2: “Which universe? Are you talking about a single maths+physics universe? (as opposed to Platonism)”
Response 2: Leading to a definition of a universe that is a single maths+physics universe, which can’t be defined consistency. Which means we end up dispensing with it and then having something closer to Platonism.
Comment 3: “You can specify which kind of MU you are talking about, as a hypothesis. (although at this point I am not even clear whether you are for or against MR.”
Response 3: I don’t specify a MU as I don’t assume such a one exists, in an encapsulated way. I specify that we hold that mathematical truths are real (exist) and show that they need to lie outside the physical universe. I think this must be the heart of the confusion—there must be a definition of MR that is different from the way I am using it. It appears as though, in your view, that having something ‘exist’ means it needs to be encapsulated either in a) a universe, or b) the physical universe so they ar combined together. (Not sure which you are assuming), whereas I had not taken MR to mean this, otherwise the work wouldn’t make sense. Apart from the terminology, I rely on the definition of the physical universe in the manuscript, ie Eq(68).
Comment 4: “I didn’t say that. The problem is that it doesn’t make any difference to our activities—we can’t tell whether the axiom of choice is true by peaking into Plato’s heaven. So it’s a huge amount of additional entities that don’t do anything, in practice.”
Response 4: Ok, we can’t tell if an axiom is true (by definition of what an axiom is) as its one of the building blocks for a particular computational logic we may choose to define, L. Now, if L is embedded in some other framework, a meta language, we may be able to make some statements about the axioms, but not from within L itself. But, for a given L, and its axioms, the truths would be statements that are formatted as theorems, which can either be proved or take the format of a Goedel sentence (ie only proved from outside L but not from within, unless the consistency of L is undermined). In other words, the mathematical truths and realities that we can ‘peek into Plato’s heaven’ to see, aren’t the axioms, of a given L (in the context of L, unless we go outside of L), they are of the format “given a and b” (where we can make ‘a’ and ‘b’ anything we want), “then c follows”. So I am stating that such a truth holds true, for my selection of ‘a’ and ‘b’, regardless of whether one is in a physical universe or any other speculated other physical universe. ie I am asserting that these mathematical truths hold in any physical universe as a meta language, as I can choose anything I want for ‘a’ and ‘b’, then logic of ‘c’ holding up in the context of the system ‘L’ I have defined, is immutable and true.
Comment 5 ”...and there isn’t a unique non contradictory MU either.”
Response 5: Fair—indeed there wouldn’t be, as there are many different systems of logic, say, I could define (different L’s, as mentioned above, for instance) that could all be internally consistent. I don’t however state that there is such an MU encapsulated. I only really explore the physical universe and whether we can bolt-on mathematical truths to it or not. Believing mathematical theorems are real truths that hold everywhere (regardless of what universe), and finding that we make the definition of the universe tricky if we definite as the physical universe + mathematical truths, I concluded that, adding mathematical all truths enters a Cantor universality paradox. However, you could for example, argue for a subset of mathematical truths (just the ones that apply to the physical universe in some way) and bolt it on to the physical universe to make a new type of MR. I haven’t seen a prescription for how this could be done though in practice. e.g. it’s not clear a given phenomenon can even have a single mathematical prescription applied to it. I could use complex numbers, or matrices, or other types of vector spaces, to achieve a similar end—the mathematical machinery isn’t necessarily unique when applied to their relevant targets in the physical universe, and yet all these mathematical theorems and the objects they operate on are different entities—so where does it end? (ie how can we prescribe how to limit the mathematical theorems that might get bolted on to the physical universe to make a new universe including them all, so as not to run into a Cantor paradox)?
Comment 6: “The argument is that the MU is either maximal and inconsistent, or non-unique., ie indefineable. Both are problems.
Plus the Occam’s razor problem.”
Response 6: If i understand correctly what MU means here, that is some set that includes at least enough mathematical objects to encounter a Cantor paradox), then I agree an MU being maximal makes it undefinable consistently. But the other case that, if it is split into smaller/limited subsets that don’t encounter that, then there are a multiplicity of them, and not a unique MU, I don’t know if this second option is a separate problem though. If I arbitrarily separate them out into different components, and I select one of them (perhaps a candidate to bolt onto the physical universe to define a phys+math universe), I can still ask if the remaining ones I left out are ‘true’ or ‘real’ and whether they are in this newly defined universe.
I don’t know of any prescription for selecting such a limited candidate subset though.
Comment 7: “It would come from leaving out one side of a contrdiction, ie. there are two smaller universes for every contradiction.”
Response 7: Ok, which amounts to defining a smaller calculus, like an ‘L’, which has only certain axioms, contains only certain structures and is limited to only certain theorems.
Comment 8: “So.,,,have you a better one?”
Response 8: I was actually going to ask you the same question, but for MR—if it appears my definition of MR is different from what you had in mind, what should I call my thing, if the naming convention is causing confusion? Essentially what I meant was the math I wrote—the words are hard to get right as people come at the problem with different background of terminology usage.
For your question here about a better term for MU, I think we can retain it but clarify the different concepts. In one case, MU could be used to mean an encapsulated system (ie a universe) of mathematical theorems. Or in one case, it could mean one of the completely consistent definable subsets of it. In another case, it could mean the physical universe + a bolt on of either all or a consistently definable subset of mathematical theorems (ie both physical + math together). Maybe we should call the latter, ‘PMU’?