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Have you perchance been reading a lot of Wittgenstein? I think you're conflating universality and objectivity. Those terms you list all have objective definiti

Have you perchance been reading a lot of Wittgenstein? I think you're conflating universality and objectivity. Those terms you list all have objective definitions, but the specific characteristics they have in any given logic may differ. That means they are not universal, but that doesn't make them non-objective. Objective typically means "mind independent". Your example of equality already demonstrates you understand equality's objective definition: you implicitly operate on the notion that "equality" means some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change. That is an informal but objective understanding of equality. What you're recognizing is that equality can have different logical properties in different contexts, where "context" can be understood as the formal language we're using, ie. it's not universal. But it's role in any given logic is always the same and not dependent on the provers mind state or his surrounding culture, ie. it is objective. Godel showed that there is no such thing as a universal logic in our current approach to formal systems, but that didn't suddenly make logic non-objective. It simply means that there is no Ur-logic that can subsume all other logics (which is why most assert that Godel ended Hilbert's program). So what logics a culture or species may use or find interesting, and the process by which they explore these systems are socially contextual, but the structures themselves and their internal consistency is not socially constructed. A culture can certainly believe a formal system they use to be logically consistent, but that's no more interesting a statement than that some cultures believed that Thor caused lightning. In other words, they could just be wrong about the consistency of their arguments.

auggierose 4y ago
> Godel showed that there is no such thing as a universal logic in our current approach to formal systems, but that didn't suddenly make logic non-objective. It simply means that there is no Ur-logic that can subsume all other logics (which is why most assert that Godel ended Hilbert's program). Could you elaborate on that? Any references?
- naasking 4y ago
  These are the implications of Godel's incompleteness theorems. No formal system expressive enough to encode arithmetic can simultaneously be both complete and prove its own consistency, because there will always be true propositions expressible in that system that cannot be proven in that system. This is why Hilbert's program to finitely axiomatize mathematics can't be completed. The "escape hatch" here is simply that not all propositions are actually interesting, so finite axiomatizations are still very useful, and we can extend the axiomatic basis as needed given satisfactory justification. This last part is the only place where social consensus sometimes comes into play (continuum hypothesis, etc). Edit: there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible...
  - auggierose 4y ago
    Ah ok, I see what you mean. This is different from how I would understand the notion Ur-Logic: Just a logical system that can express anything you want, given you are free to add axioms. Obviously there are a few choices for that.
  - qazxcvbnm 4y ago
    > there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible... Could you elaborate on what such a formalism where infinite structures are not expressible might look like? It sounds an intriguing notion, though I'm unable to quite get at it.
    - naasking 4y ago
      Bendegem's paper is a good introduction and overview: A Defense of Strict Finitism, http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf
prmph 4y ago
Wow, did you read my comment carefully at all? This is what I said: > This is not to deny that there is an objective reality... So I do agree that object reality exists, contrary to what you imply in your response. However, there is no word in human language that has an objective definition. If you think this is so, maybe you have not thought or read enough about the subject. On the subject of equality, you said: > Your example of equality already demonstrates you understand equality's objective definition: you implicitly operate on the notion that "equality" means some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change. You're making my point. Consider the vagueness of what you have written: "...some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change." That doesn't sound very rigorous to me, and in fact I could argue this is not what I understand by "equality" at all. Now you could make the point the definition can be made rigorous for a very specific context, but this is still not 100% possible. To show this, you can attempt to give me a specific definition of equality of a specific context, and I will show that it still admits of multiple possible meanings.

auggierose 4y ago

> Godel showed that there is no such thing as a universal logic in our current approach to formal systems, but that didn't suddenly make logic non-objective. It simply means that there is no Ur-logic that can subsume all other logics (which is why most assert that Godel ended Hilbert's program). Could you elaborate on that? Any references?

naasking 4y ago
These are the implications of Godel's incompleteness theorems. No formal system expressive enough to encode arithmetic can simultaneously be both complete and prove its own consistency, because there will always be true propositions expressible in that system that cannot be proven in that system. This is why Hilbert's program to finitely axiomatize mathematics can't be completed. The "escape hatch" here is simply that not all propositions are actually interesting, so finite axiomatizations are still very useful, and we can extend the axiomatic basis as needed given satisfactory justification. This last part is the only place where social consensus sometimes comes into play (continuum hypothesis, etc). Edit: there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible...
- auggierose 4y ago
  Ah ok, I see what you mean. This is different from how I would understand the notion Ur-Logic: Just a logical system that can express anything you want, given you are free to add axioms. Obviously there are a few choices for that.
- qazxcvbnm 4y ago
  > there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible... Could you elaborate on what such a formalism where infinite structures are not expressible might look like? It sounds an intriguing notion, though I'm unable to quite get at it.
  - naasking 4y ago
    Bendegem's paper is a good introduction and overview: A Defense of Strict Finitism, http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf

prmph 4y ago

Wow, did you read my comment carefully at all? This is what I said: > This is not to deny that there is an objective reality... So I do agree that object reality exists, contrary to what you imply in your response. However, there is no word in human language that has an objective definition. If you think this is so, maybe you have not thought or read enough about the subject. On the subject of equality, you said: > Your example of equality already demonstrates you understand equality's objective definition: you implicitly operate on the notion that "equality" means some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change. You're making my point. Consider the vagueness of what you have written: "...some form of equivalence, some ability to substitute B for C in a specific context that results in no observable/expressible change." That doesn't sound very rigorous to me, and in fact I could argue this is not what I understand by "equality" at all. Now you could make the point the definition can be made rigorous for a very specific context, but this is still not 100% possible. To show this, you can attempt to give me a specific definition of equality of a specific context, and I will show that it still admits of multiple possible meanings.