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But what this doesn't get at is that the very system of logic we use to make proofs is a social construct. Other cultures have had other systems of logic, and c

But what this doesn't get at is that the very system of logic we use to make proofs is a social construct. Other cultures have had other systems of logic, and called valid arguments that we wouldn't today precisely because they were using a different system of logic. So the very foundations of mathematics, the logic we use behind our proofs, is inherently a social construct that arose out of Greek philosophy as it was adapted in the west.

naasking 4y ago
> But what this doesn't get at is that the very system of logic we use to make proofs is a social construct. Other cultures have had other systems of logic, and called valid arguments that we wouldn't today precisely because they were using a different system of logic. I think you need to separate the process of developing mathematics, from the self-consistency and validity of the logical argument or mathematical structure itself. The process is social but the validity and mathematical structure itself is not. Computer science has led to an explosion in different logics. We've never had more logics/formal systems than we do today, but whether any given formal system actually is consistent is not dependent on consensus, it is a fact, either true or false, completely independent of consensus.
- musingsole 4y ago
  > whether any given formal system actually is consistent is not dependent on consensus, it is a fact, either true or false, completely independent of consensus. The definition of "consistent" seems completely entangled with a given social group's ideas of "rational". You might imply that our word for it hints at a Platonic ideal of "consistent", but if that's true, then you're caught in an infinite cascade of which nuances of meaning between the Platonic Ideal and our concrete reality are actually reflections of the truth or corruption of it.
  - prmph 4y ago
    Exactly the point I was making. Many do not seems to see the fundamental issue at play here, and another way to think of them is what you have hinted at: the role of language. There is no objective way to nail down the meaning of words, like "consistent", "proof", "equal", etc. Suppose one wanted a maximally rigorous definition of "equal". Does it mean two things that cause people to think of the same thing when they are mentioned? Does it mean two things that occupy the same position in space at all times? It is actually a difficult concept to define rigorously. This is not to deny that there is an objective reality. But that reality is highly contextual and multi-faceted. We cannot be 100% exact in defining that reality using language (even a math language), and this is where the social nature of that reality becomes apparent. The role of proofs are in creating, as far as possible, as rigorous a shared context for the reality being described.
    - auggierose 4y ago
      Equality is actually quite easy to axiomatise in most logics, here in my favourite logic: 1) x = x 2) x = y => P[x] => P[y] In my opinion, there is a mathematical reality, which is shared by everyone, even by those who don't believe in it :-) For example, a logical system exists in that reality, and you can either derive a theorem in that system or not in this reality. I don't think it is possible that there is a third possibility. I don't think it is possible that I have a different reality from you in that respect. This reality is not socially constructed, it just is. Intuitionism will tell you that because you don't know if a certain theorem is derivable, it is in some sort of hybrid state until we know for sure via a intuitionistic proof or a counter example. I think that is bullocks. Either there is a proof or not. Either there is a counter example, or not. Beyond that, extending this mathematical reality, there is a wider, not as easily accessible reality. We can try to understand that reality by modelling it via certain assumptions, and then applying our mathematical reality to those assumptions. I believe the mathematical conclusions we draw from this will be real to the extent that the assumptions are true; but of course you cannot ever be sure about those assumptions, and so you cannot be sure about the conclusions. But if you notice that your conclusions do not hold, you need to challenge your assumptions, not your mathematics.
      - 4y ago
  - auggierose 4y ago
    There is a simple definition for what consistent means, which naasking is referring to: is it impossible to derive "false" purely by applying the rules of the logical system?
    - prmph 4y ago
      "False" is meaningless in a logic system that relies on probabilities in stead or true/false. Even in our commonly used logic system, what does false actually mean?
      - auggierose 4y ago
        I can point you to my favourite logic ;-) Here false is defined as false = (∀x. x) So in this case, it derives its meaning from whatever the meaning of the ∀ operator is. Its meaning is the result of applying the ∀ operator to the identity operation. Nobody is questioning that there are different possibilities for logical systems, and different possible semantics for them. The choice of logical system is up to you, and ultimately, part of your assumptions. A proof assistant uses a fixed logical system, and your mathematics needs to fit into this system to be expressible in the proof assistant. Apart from my favourite logic, there are several well-known logical systems commonly assumed to be able to represent all of mathematics more or less faithfully: first-order logic, simply-typed higher-order logic, and dependent type theory. What I like about my favourite logic is that I believe that all other logics, even quantum logics, can be expressed in it in a straightforward way. This has the advantage that once you expressed your logic in this way, you get the meaning of it for free. Of course, you still need to make sure that this is the intended meaning.
        musingsole 4y ago
        You draw lines and insist there's no space between them.
        auggierose 4y ago
        Not sure what you are trying to say. Care to elaborate?

naasking 4y ago

> But what this doesn't get at is that the very system of logic we use to make proofs is a social construct. Other cultures have had other systems of logic, and called valid arguments that we wouldn't today precisely because they were using a different system of logic. I think you need to separate the process of developing mathematics, from the self-consistency and validity of the logical argument or mathematical structure itself. The process is social but the validity and mathematical structure itself is not. Computer science has led to an explosion in different logics. We've never had more logics/formal systems than we do today, but whether any given formal system actually is consistent is not dependent on consensus, it is a fact, either true or false, completely independent of consensus.

musingsole 4y ago
> whether any given formal system actually is consistent is not dependent on consensus, it is a fact, either true or false, completely independent of consensus. The definition of "consistent" seems completely entangled with a given social group's ideas of "rational". You might imply that our word for it hints at a Platonic ideal of "consistent", but if that's true, then you're caught in an infinite cascade of which nuances of meaning between the Platonic Ideal and our concrete reality are actually reflections of the truth or corruption of it.
- prmph 4y ago
  Exactly the point I was making. Many do not seems to see the fundamental issue at play here, and another way to think of them is what you have hinted at: the role of language. There is no objective way to nail down the meaning of words, like "consistent", "proof", "equal", etc. Suppose one wanted a maximally rigorous definition of "equal". Does it mean two things that cause people to think of the same thing when they are mentioned? Does it mean two things that occupy the same position in space at all times? It is actually a difficult concept to define rigorously. This is not to deny that there is an objective reality. But that reality is highly contextual and multi-faceted. We cannot be 100% exact in defining that reality using language (even a math language), and this is where the social nature of that reality becomes apparent. The role of proofs are in creating, as far as possible, as rigorous a shared context for the reality being described.
  - auggierose 4y ago
    Equality is actually quite easy to axiomatise in most logics, here in my favourite logic: 1) x = x 2) x = y => P[x] => P[y] In my opinion, there is a mathematical reality, which is shared by everyone, even by those who don't believe in it :-) For example, a logical system exists in that reality, and you can either derive a theorem in that system or not in this reality. I don't think it is possible that there is a third possibility. I don't think it is possible that I have a different reality from you in that respect. This reality is not socially constructed, it just is. Intuitionism will tell you that because you don't know if a certain theorem is derivable, it is in some sort of hybrid state until we know for sure via a intuitionistic proof or a counter example. I think that is bullocks. Either there is a proof or not. Either there is a counter example, or not. Beyond that, extending this mathematical reality, there is a wider, not as easily accessible reality. We can try to understand that reality by modelling it via certain assumptions, and then applying our mathematical reality to those assumptions. I believe the mathematical conclusions we draw from this will be real to the extent that the assumptions are true; but of course you cannot ever be sure about those assumptions, and so you cannot be sure about the conclusions. But if you notice that your conclusions do not hold, you need to challenge your assumptions, not your mathematics.
    - 4y ago
- auggierose 4y ago
  There is a simple definition for what consistent means, which naasking is referring to: is it impossible to derive "false" purely by applying the rules of the logical system?
  - prmph 4y ago
    "False" is meaningless in a logic system that relies on probabilities in stead or true/false. Even in our commonly used logic system, what does false actually mean?
    - auggierose 4y ago
      I can point you to my favourite logic ;-) Here false is defined as false = (∀x. x) So in this case, it derives its meaning from whatever the meaning of the ∀ operator is. Its meaning is the result of applying the ∀ operator to the identity operation. Nobody is questioning that there are different possibilities for logical systems, and different possible semantics for them. The choice of logical system is up to you, and ultimately, part of your assumptions. A proof assistant uses a fixed logical system, and your mathematics needs to fit into this system to be expressible in the proof assistant. Apart from my favourite logic, there are several well-known logical systems commonly assumed to be able to represent all of mathematics more or less faithfully: first-order logic, simply-typed higher-order logic, and dependent type theory. What I like about my favourite logic is that I believe that all other logics, even quantum logics, can be expressed in it in a straightforward way. This has the advantage that once you expressed your logic in this way, you get the meaning of it for free. Of course, you still need to make sure that this is the intended meaning.
      - musingsole 4y ago
        You draw lines and insist there's no space between them.
        auggierose 4y ago
        Not sure what you are trying to say. Care to elaborate?