Regarding your original formulation, I think you could phrase things a little more simply. For example: “For a mathematical claim to be true, we require two conditions. Firstly, the axioms of the claim should agree with our own accepted axioms, and our axioms should be reasonable. Secondly, the claim should follow from those axioms.” As far as I can tell, this is basically what you’re saying, although what you mean by the reasonableness of our axioms is unclear to me.
Regarding the existence of mathematical entities, you’ve seemed to answer in the negative. But I don’t see this as following from your original framework. That framework says nothing about the identity of the mathematical structures in and of themselves. That framework only tells you how the math works, not what it is. Although perhaps you’re saying that the math structures have no identity in and of themselves, existing solely in this framework of claim verification. But in that case, the resolution is tautological, and I’m not sure it really gets to the heart of that question.
In looking at the discovery versus invention of math, you support the invention. But this is essentially a reconfiguration of the previous problem. If we don’t know precisely the identity of the math, the difference between its invention or its discovery is moot.
I disagree about your negative conclusion of math making new predictions a priori. There is the following underlying problem. The conditions of the reality in general do not exactly match the conditions of the math, or at least, this is not verifiable in general. Hence you can never be sure that your isomorphism between math and reality is strictly correct. But that means that a priori mathematical reasoning is the de facto standard (in general). Obviously there are some special cases like adding apples or rocks together which seems to be fully correspondent, but in most cases, the isomorphism may be unverifiable. That’s why it’s amazing that it works.
Regarding the evidence for the truthfulness of math statements… this “truthfulness” just follows by construction from within the original framework. Not sure what you were getting at in that section.
Universes with other math systems—I like this section of yours the most, I think it at least correctly identifies the non triviality of that possibility. A system where non-trivially 2+2=6 would have to correspond to a bit of a different reality, but that reality would also presumably be self-consistent. But if it is self-consistent, then that statement would have to make sense within the system itself. Therefore you escape any problems or contradictions and it loses its idea of being special or strange.
Anyways hope this feedback might be helpful for you and I apologize if I’m misinterpreting you here, as I’ve done a lot of synthesis in this comment.
Thanks for the feedback. I’ll reply to your concerns as best I can.
As far as I can tell, this is basically what you’re saying, although what you mean by the reasonableness of our axioms is unclear to me.
I didn’t require the axioms to be reasonable in this approach, except, of course, to the extent that their reasonableness causes people to generally accept them in common usage.
Although perhaps you’re saying that the math structures have no identity in and of themselves, existing solely in this framework of claim verification. But in that case, the resolution is tautological, and I’m not sure it really gets to the heart of that question.
That is indeed what I’m saying, but I disagree that it’s tautological. To the extent that my framework handles difficult problems and paradoxes in a satisfactory way, that is its non-tautological substantiation, as it shows how you don’t need to appeal to concepts outside of what I have reduced math to.
I disagree about your negative conclusion of math making new predictions a priori. There is the following underlying problem. The conditions of the reality in general do not exactly match the conditions of the math, or at least, this is not verifiable in general. Hence you can never be sure that your isomorphism between math and reality is strictly correct. But that means that a priori mathematical reasoning is the de facto standard (in general). Obviously there are some special cases like adding apples or rocks together which seems to be fully correspondent, but in most cases, the isomorphism may be unverifiable.
I mostly agree, but refer back to the causal diagram. As a standard Bayesian rule, you will never have 100% certainty on any of your premises or conclusions. However, failure of the predicted causal implication to hold (“adding two rocks to two rocks will yield four rocks”) needn’t have the same impact on your degree of belief in each of its causal parents. You can do a lot more to verify your math than to verify the isomorphism to something physical.
If the isomorphism has a lot of evidence favoring it, then the math can tell you surprising things about particular regions of the domain of supposed applicability, which turn out to be true. This is the essence of science and engineering. My point here is only that the math’s applicability to the universe always depends on the empirical validity of the isomorphism, which you might miss if you view the output of math as being the critical step in an insight.
That’s why it’s amazing that it works.
I think the amazingness will eventually be demystified by a common factor that caused both our use of math and the universe’s frequent close isomorphisms thereto.
Regarding the evidence for the truthfulness of math statements… this “truthfulness” just follows by construction from within the original framework. Not sure what you were getting at in that section.
Yes, and the framework can be relevant or irrelevant to physical systems; people are more likely to be referring to axiom sets that are relevant (have an isomorphism) to physical systems.
Regarding your original formulation, I think you could phrase things a little more simply. For example: “For a mathematical claim to be true, we require two conditions. Firstly, the axioms of the claim should agree with our own accepted axioms, and our axioms should be reasonable. Secondly, the claim should follow from those axioms.” As far as I can tell, this is basically what you’re saying, although what you mean by the reasonableness of our axioms is unclear to me.
Regarding the existence of mathematical entities, you’ve seemed to answer in the negative. But I don’t see this as following from your original framework. That framework says nothing about the identity of the mathematical structures in and of themselves. That framework only tells you how the math works, not what it is. Although perhaps you’re saying that the math structures have no identity in and of themselves, existing solely in this framework of claim verification. But in that case, the resolution is tautological, and I’m not sure it really gets to the heart of that question.
In looking at the discovery versus invention of math, you support the invention. But this is essentially a reconfiguration of the previous problem. If we don’t know precisely the identity of the math, the difference between its invention or its discovery is moot.
I disagree about your negative conclusion of math making new predictions a priori. There is the following underlying problem. The conditions of the reality in general do not exactly match the conditions of the math, or at least, this is not verifiable in general. Hence you can never be sure that your isomorphism between math and reality is strictly correct. But that means that a priori mathematical reasoning is the de facto standard (in general). Obviously there are some special cases like adding apples or rocks together which seems to be fully correspondent, but in most cases, the isomorphism may be unverifiable. That’s why it’s amazing that it works.
Regarding the evidence for the truthfulness of math statements… this “truthfulness” just follows by construction from within the original framework. Not sure what you were getting at in that section.
Universes with other math systems—I like this section of yours the most, I think it at least correctly identifies the non triviality of that possibility. A system where non-trivially 2+2=6 would have to correspond to a bit of a different reality, but that reality would also presumably be self-consistent. But if it is self-consistent, then that statement would have to make sense within the system itself. Therefore you escape any problems or contradictions and it loses its idea of being special or strange.
Anyways hope this feedback might be helpful for you and I apologize if I’m misinterpreting you here, as I’ve done a lot of synthesis in this comment.
Thanks for the feedback. I’ll reply to your concerns as best I can.
I didn’t require the axioms to be reasonable in this approach, except, of course, to the extent that their reasonableness causes people to generally accept them in common usage.
That is indeed what I’m saying, but I disagree that it’s tautological. To the extent that my framework handles difficult problems and paradoxes in a satisfactory way, that is its non-tautological substantiation, as it shows how you don’t need to appeal to concepts outside of what I have reduced math to.
I mostly agree, but refer back to the causal diagram. As a standard Bayesian rule, you will never have 100% certainty on any of your premises or conclusions. However, failure of the predicted causal implication to hold (“adding two rocks to two rocks will yield four rocks”) needn’t have the same impact on your degree of belief in each of its causal parents. You can do a lot more to verify your math than to verify the isomorphism to something physical.
If the isomorphism has a lot of evidence favoring it, then the math can tell you surprising things about particular regions of the domain of supposed applicability, which turn out to be true. This is the essence of science and engineering. My point here is only that the math’s applicability to the universe always depends on the empirical validity of the isomorphism, which you might miss if you view the output of math as being the critical step in an insight.
I think the amazingness will eventually be demystified by a common factor that caused both our use of math and the universe’s frequent close isomorphisms thereto.
Yes, and the framework can be relevant or irrelevant to physical systems; people are more likely to be referring to axiom sets that are relevant (have an isomorphism) to physical systems.