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’?
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’?