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> Mathematical truths are defined to be true Perhaps a nitpick, but: axioms are defined to be true. Mathematics, roughly speaking, is the business of exploring

> Mathematical truths are defined to be true Perhaps a nitpick, but: axioms are defined to be true. Mathematics, roughly speaking, is the business of exploring the consequences of chosen sets of axioms. Sometimes published mathematical theorems turn out to be wrong.

jjgreen 6y ago
Do you think? I'd say that axioms were just taken; there's no commitment to their truth (since there is no need for it, and mathematics likes economy).
- qznc 6y ago
  Axiom are just taken but over time we picked the most useful ones. This is why they map well to the physical world. The baffling thing is how few and simple axioms we need for them to be useful.
- amelius 6y ago
  It also doesn't make sense to define axioms as true. What if later it turns out that the axioms were a mutual contradiction?
  - jjgreen 6y ago
    Truth is tricky. If I state "The sky is blue", I can look out of the window, the sky is blue, my statement corresponds to the state of the world, my statement is true, right? But in mathematics there is no window, everything is internal, abstract, so what does "truth" mean when there is no state of the world with which to compare? More importantly, would the mathematics be any different if the axioms are declared to be true? No, it wouldn't make any difference at all. So let's leave "truth" to the physicists and philosophers.
    - chousuke 6y ago
      This is perhaps not all that relevant to your point, but "the sky is blue" so often being used as an example of an obvious truth is amusing to me, because very often the sky is not all that blue at all. If I look outside right now, the sky is probably better described as cyan; quite obviously a different colour from the blue exercise mat that happens to be on my floor. The sky can also be described by many of the other colours depending on weather or the time of day.
    - ukj 6y ago
      That's the first error of the objectivists. Mistaking "blue" as a property of the sky. Color happens in your head. That is why trichromacy, tetrachromacy, pentachromacy, dodecachromacy etc. is a thing.
    - RandoHolmes 6y ago
      It's almost as if defining anything as a "mathematical truth" is definitionally wrong and anyone having done so should be called out as a liar. Only the person who responded to the person who made such a statement were more polite about it, hence this "discussion".
  - mundo 6y ago
    This might be a good time to look up the definition of axiom: "a statement accepted as true as the basis for argument or inference".
    - amelius 6y ago
      Yes, but I think that's "accepted" in the sense of "until we run into a contradiction", at which point you have to start over with new axioms. I.e. it turned out your axioms weren't all true. I think that in the definition you gave "accepted" should be replaced by "assumed".
      - mundo 6y ago
        "John can dig a hole ten feet deep in an hour. How long would it take him to dig a hole five feet deep?" The first sentence of that problem is an axiom. It is absolutely 100% true, within the context of that story problem. It could contradict another axiom, but it can't turn out to be false, because it's not a statement about the real world.
- fenomas 6y ago
  In math "true" is essentially just a name for a symbol. Saying that an axiom is true is like saying that 3 comes after 2 - it doesn't imply any kind of truth in the usual "consistent with reality" sense.
fenomas 6y ago
I don't follow what you're taking issue with. Naturally by "mathematical truths" I meant axioms and statements that are reducible to axioms - there isn't really anything else for it to mean, after all. On the "sometimes published theorems are wrong" part you've lost me entirely.
- MaxBarraclough 6y ago
  > by "mathematical truths" I meant axioms and statements that are reducible to axioms That distinction is important, and it seemed to me you'd skimmed over it. Axioms are directly defined to be true for the purposes of mathematical inquiry, and as such can't really be wrong. At worst, a set of axioms can be inconsistent. Emergent properties are different; they're not directly defined to be true. This is of practical consequence, as mathematicians sometimes get them wrong.
RandoHolmes 6y ago
Thank you, I took the same issue you did when reading this. There are not "truths" in mathematics, it's literally a system of logic with definitions at the bottom (axioms) and then a following of the results based upon the agreed upon logic.

jjgreen 6y ago

Do you think? I'd say that axioms were just taken; there's no commitment to their truth (since there is no need for it, and mathematics likes economy).

qznc 6y ago
Axiom are just taken but over time we picked the most useful ones. This is why they map well to the physical world. The baffling thing is how few and simple axioms we need for them to be useful.
amelius 6y ago
It also doesn't make sense to define axioms as true. What if later it turns out that the axioms were a mutual contradiction?
- jjgreen 6y ago
  Truth is tricky. If I state "The sky is blue", I can look out of the window, the sky is blue, my statement corresponds to the state of the world, my statement is true, right? But in mathematics there is no window, everything is internal, abstract, so what does "truth" mean when there is no state of the world with which to compare? More importantly, would the mathematics be any different if the axioms are declared to be true? No, it wouldn't make any difference at all. So let's leave "truth" to the physicists and philosophers.
  - chousuke 6y ago
    This is perhaps not all that relevant to your point, but "the sky is blue" so often being used as an example of an obvious truth is amusing to me, because very often the sky is not all that blue at all. If I look outside right now, the sky is probably better described as cyan; quite obviously a different colour from the blue exercise mat that happens to be on my floor. The sky can also be described by many of the other colours depending on weather or the time of day.
  - ukj 6y ago
    That's the first error of the objectivists. Mistaking "blue" as a property of the sky. Color happens in your head. That is why trichromacy, tetrachromacy, pentachromacy, dodecachromacy etc. is a thing.
  - RandoHolmes 6y ago
    It's almost as if defining anything as a "mathematical truth" is definitionally wrong and anyone having done so should be called out as a liar. Only the person who responded to the person who made such a statement were more polite about it, hence this "discussion".
- mundo 6y ago
  This might be a good time to look up the definition of axiom: "a statement accepted as true as the basis for argument or inference".
  - amelius 6y ago
    Yes, but I think that's "accepted" in the sense of "until we run into a contradiction", at which point you have to start over with new axioms. I.e. it turned out your axioms weren't all true. I think that in the definition you gave "accepted" should be replaced by "assumed".
    - mundo 6y ago
      "John can dig a hole ten feet deep in an hour. How long would it take him to dig a hole five feet deep?" The first sentence of that problem is an axiom. It is absolutely 100% true, within the context of that story problem. It could contradict another axiom, but it can't turn out to be false, because it's not a statement about the real world.
fenomas 6y ago
In math "true" is essentially just a name for a symbol. Saying that an axiom is true is like saying that 3 comes after 2 - it doesn't imply any kind of truth in the usual "consistent with reality" sense.

fenomas 6y ago

I don't follow what you're taking issue with. Naturally by "mathematical truths" I meant axioms and statements that are reducible to axioms - there isn't really anything else for it to mean, after all. On the "sometimes published theorems are wrong" part you've lost me entirely.

MaxBarraclough 6y ago
> by "mathematical truths" I meant axioms and statements that are reducible to axioms That distinction is important, and it seemed to me you'd skimmed over it. Axioms are directly defined to be true for the purposes of mathematical inquiry, and as such can't really be wrong. At worst, a set of axioms can be inconsistent. Emergent properties are different; they're not directly defined to be true. This is of practical consequence, as mathematicians sometimes get them wrong.

RandoHolmes 6y ago

Thank you, I took the same issue you did when reading this. There are not "truths" in mathematics, it's literally a system of logic with definitions at the bottom (axioms) and then a following of the results based upon the agreed upon logic.