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> The equation sign is used ambiguously. (for definition and for equality) Definition and equality are really the same thing. A definition is just an introduct

> The equation sign is used ambiguously. (for definition and for equality) Definition and equality are really the same thing. A definition is just an introduction of a term/variable, and an equality giving it a value. So the only difference is if a term in the equation hasn't been introduced yet. > The bracketing placement of symbols for absolute value, is not a matching pair, thus is ambiguous (when nested or sequential). How was it not a matching pair? I only see two |'s in each equation so they have to pair up. What other way is there to interpret it? > the notation for differentials “dy/dx” rapes the division notation. The differential notion has no position in math regarded as a computer language, as implied by the math philosophies of formalism and logicism. I believe the original reasoning for notating derivatives as dy/dx is that it is a limit of a faction and so approaches a fraction of infinitesimals. with that background the division notation is simply extended to a new class of numbers. I am not sure what the second sentence is trying to say. Also pretty much all of the issues could be solved with gratuitous parenthesis. So we have a solution to get a formal grammar; we use the current notation because it is faster to read and write. So without any concrete proposals, this critique has very little content.

deleted 9y ago
[deleted]
yorwba 9y ago
Absolute value is ambiguous if you have more than one occurence, like |a|b|c| could be (|a|)b(|c|) or |a(|b|)c|. But as you said, gratuitous parentheses solve this problem. A good typesetting program will also let you display the outer bars larger than the inner ones, so you can match them up even without parentheses
- thethirdone 9y ago
  > Absolute value is ambiguous if you have more than one occurence, like |a|b|c| could be (|a|)b(|c|) or |a(|b|)c|. Yeah, I get the issue in general, but the quote I referenced was specifically referring to the equations from Wikipedia. My thought is that pretty much any mathematician would not choose to use absolute value bars in a case where it would be ambiguous. For example (|a|)b(|c|) would be instead written |ac|b and |a(|b|)c| would be written |abc|. And so unless it the equations are specifically about showing properties of absolute value (in which case just use parenthesis), I don't see a case where the ambiguity would pop up in real discourse.
  - chongli 9y ago
    For example (|a|)b(|c|) would be instead written |ac|b That changes the meaning. You're now relying on multiplication being commutative which is not true for all sets.
    - netheril96 9y ago
      It doesn't matter in practice. These two bars denote either absolute value, or norm, or determinant, all of which are real or complex numbers. People surely have the freedom to define a non-commutative multiplication with real/complex scalars, but have you ever seen one?
    - thethirdone 9y ago
      > That changes the meaning. You're now relying on multiplication being commutative which is not true for all sets As I said at the end of my last comment: I don't see a case where the ambiguity would pop up in real discourse. If you are dealing with non-commutative objects you can deal with a few more parenthesis. Just like non-associative objects result in many more.
theoh 9y ago
> Definition and equality are really the same thing. A definition is just an introduction of a term/variable, and an equality giving it a value. So the only difference is if a term in the equation hasn't been introduced yet. In CS we talk about l-values and r-values. Only an l-value can be assigned to, but you can compare r-values for equality. So there is a solid distinction there between types of expression which is upheld in programming languages.
- contravariant 9y ago
  The main difference between math and programming languages is that it's perfectly fine in math to define l-values. For example i^2 = -1 is the generally accepted definition of the imaginary unit.
  - theoh 9y ago
    It's r-values that cannot be assigned to. What's really missing from most conventional math, I think, is the notion of conditional expressions based on equality testing. "=" is usually only used in a declarative way, to declare that two expressions are equal. See n+k patterns in Haskell for a programming exception to the rule you mention.
    - theoh 9y ago
      Actually, though, set-builder notation can use the equals sign to express a condition. e.g. the examples here https://en.wikipedia.org/wiki/Set-builder_notation https://en.wikipedia.org/wiki/Set-builder_notation. Maybe there are other examples in math.
tsilva 9y ago
> Definition and equality are really the same thing. [...] So the only difference is [...]. An interesting way of justifying how two things are really the same thing, that is, by acknowledging their difference. The point is that the pointed ambiguity (the difference) can be quite significant, and these become apparent when studying these sentences in a formal logic. Hence, the significance of the ambiguity. This (and the article under discussion) just echoes the idea that mathematical sentences and proofs are considered "rigorous" but "informal", where "formal" is the domain of logic.

deleted 9y ago

[deleted]

yorwba 9y ago

Absolute value is ambiguous if you have more than one occurence, like |a|b|c| could be (|a|)b(|c|) or |a(|b|)c|. But as you said, gratuitous parentheses solve this problem. A good typesetting program will also let you display the outer bars larger than the inner ones, so you can match them up even without parentheses

thethirdone 9y ago
> Absolute value is ambiguous if you have more than one occurence, like |a|b|c| could be (|a|)b(|c|) or |a(|b|)c|. Yeah, I get the issue in general, but the quote I referenced was specifically referring to the equations from Wikipedia. My thought is that pretty much any mathematician would not choose to use absolute value bars in a case where it would be ambiguous. For example (|a|)b(|c|) would be instead written |ac|b and |a(|b|)c| would be written |abc|. And so unless it the equations are specifically about showing properties of absolute value (in which case just use parenthesis), I don't see a case where the ambiguity would pop up in real discourse.
- chongli 9y ago
  For example (|a|)b(|c|) would be instead written |ac|b That changes the meaning. You're now relying on multiplication being commutative which is not true for all sets.
  - netheril96 9y ago
    It doesn't matter in practice. These two bars denote either absolute value, or norm, or determinant, all of which are real or complex numbers. People surely have the freedom to define a non-commutative multiplication with real/complex scalars, but have you ever seen one?
  - thethirdone 9y ago
    > That changes the meaning. You're now relying on multiplication being commutative which is not true for all sets As I said at the end of my last comment: I don't see a case where the ambiguity would pop up in real discourse. If you are dealing with non-commutative objects you can deal with a few more parenthesis. Just like non-associative objects result in many more.

theoh 9y ago

> Definition and equality are really the same thing. A definition is just an introduction of a term/variable, and an equality giving it a value. So the only difference is if a term in the equation hasn't been introduced yet. In CS we talk about l-values and r-values. Only an l-value can be assigned to, but you can compare r-values for equality. So there is a solid distinction there between types of expression which is upheld in programming languages.

contravariant 9y ago
The main difference between math and programming languages is that it's perfectly fine in math to define l-values. For example i^2 = -1 is the generally accepted definition of the imaginary unit.
- theoh 9y ago
  It's r-values that cannot be assigned to. What's really missing from most conventional math, I think, is the notion of conditional expressions based on equality testing. "=" is usually only used in a declarative way, to declare that two expressions are equal. See n+k patterns in Haskell for a programming exception to the rule you mention.
  - theoh 9y ago
    Actually, though, set-builder notation can use the equals sign to express a condition. e.g. the examples here https://en.wikipedia.org/wiki/Set-builder_notation https://en.wikipedia.org/wiki/Set-builder_notation. Maybe there are other examples in math.

tsilva 9y ago

> Definition and equality are really the same thing. [...] So the only difference is [...]. An interesting way of justifying how two things are really the same thing, that is, by acknowledging their difference. The point is that the pointed ambiguity (the difference) can be quite significant, and these become apparent when studying these sentences in a formal logic. Hence, the significance of the ambiguity. This (and the article under discussion) just echoes the idea that mathematical sentences and proofs are considered "rigorous" but "informal", where "formal" is the domain of logic.