3 ms·

Elements is formal, right?

hollerith 2y ago
Pretty much. According to someone on Stack Exchange, "The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof." But making mistakes in a formal treatment of a subject does not negate the fact that it is a formal treatment of the subject IMHO, and AFAICR none of the theorems in Elements is wrong; i.e., Elements is formal enough to have avoided a mistake even though the axioms listed weren't all the axiom that are actually needed to support the theorems. 18th Century European math was much more potent than ancient Greek math, and although parts of it like algebra and geometry were, for a long time, most of it was not understood at a formal or rigorous level for a long time even if we accept the level of rigor found in Elements.
- koolala 2y ago
  Isn't how formal or rigorous something is just a social convention? Grammer Nazi's used to make online speech be formal with perfect rigor. Isn't it all relative to what your society defines?
  - chongli 2y ago
    No. That's the colloquial definition of formal. In mathematics, the word formal refers to something more specific: one or more statements written using a set of symbols which have fully-defined rules for mechanically transforming them into another form. A formal proof is then one which proceeds by a series of these mechanical steps beginning with one or more premises and ending with a conclusion (or goal).
    - koolala 2y ago
      Both are a formality based on whats in fashion. I like the Axiom of Choice and not taking Math or words as literal or biblical truth.
      - deleted 2y ago
        [deleted]
      - chongli 2y ago
        If you're a formalist in philosophy of math, then math is neither true nor false, it's merely a bunch of meaningless symbols you transform via mechanical rules.
  - nyssos 2y ago
    To an extent. A truly completely formal proof, as in symbol manipulation according to the rules of some formal system, no. It's valid or it isn't. But no one actually works like this. There are varying degrees of "semiformality" and what is and isn't acceptable is ultimately a convention, and varies between subfields - but even the laxest mathematicians are still about as careful as the most rigorous physicists.
gowld 2y ago
Elements is mostly formal, but it's also concrete and visual. Euclid developed arithmetic and algebra through constructive geometry, which relies on our visual intuition to solve problems. Non-concrete problems were totally out of scope. Even curved surfaces (denying the parallel postulate) were byond Euclid. Notably, Elements didn't have imaginary or transcendental numbers. Euclid made no attempt to unify line lengths and arc lengths, and had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers.
- paulddraper 2y ago
  > Elements didn't have imaginary or transcendental numbers Elements has pi.* *It proves the ratio of a circle's area to the square of its diameter is constant.
  - gowld 2y ago
    But doesn't say how large that constant value is.
- Someone 2y ago
  > Euclid […] had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers Did he even know those gaps existed? Euclid lived around 300BC. The problem of squaring the circle had been proposed around two centuries before that (https://en.wikipedia.org/wiki/Anaxagoras#Mathematics https://en.wikipedia.org/wiki/Anaxagoras#Mathematics), but I don’t think people even considered it to be impossible by that time.
Koshkin 2y ago
Incidentally, here's one beautiful edition: https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

Pretty much. According to someone on Stack Exchange, "The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof." But making mistakes in a formal treatment of a subject does not negate the fact that it is a formal treatment of the subject IMHO, and AFAICR none of the theorems in Elements is wrong; i.e., Elements is formal enough to have avoided a mistake even though the axioms listed weren't all the axiom that are actually needed to support the theorems. 18th Century European math was much more potent than ancient Greek math, and although parts of it like algebra and geometry were, for a long time, most of it was not understood at a formal or rigorous level for a long time even if we accept the level of rigor found in Elements.

koolala 2y ago
Isn't how formal or rigorous something is just a social convention? Grammer Nazi's used to make online speech be formal with perfect rigor. Isn't it all relative to what your society defines?
- chongli 2y ago
  No. That's the colloquial definition of formal. In mathematics, the word formal refers to something more specific: one or more statements written using a set of symbols which have fully-defined rules for mechanically transforming them into another form. A formal proof is then one which proceeds by a series of these mechanical steps beginning with one or more premises and ending with a conclusion (or goal).
  - koolala 2y ago
    Both are a formality based on whats in fashion. I like the Axiom of Choice and not taking Math or words as literal or biblical truth.
    - deleted 2y ago
      [deleted]
    - chongli 2y ago
      If you're a formalist in philosophy of math, then math is neither true nor false, it's merely a bunch of meaningless symbols you transform via mechanical rules.
- nyssos 2y ago
  To an extent. A truly completely formal proof, as in symbol manipulation according to the rules of some formal system, no. It's valid or it isn't. But no one actually works like this. There are varying degrees of "semiformality" and what is and isn't acceptable is ultimately a convention, and varies between subfields - but even the laxest mathematicians are still about as careful as the most rigorous physicists.

gowld 2y ago

Elements is mostly formal, but it's also concrete and visual. Euclid developed arithmetic and algebra through constructive geometry, which relies on our visual intuition to solve problems. Non-concrete problems were totally out of scope. Even curved surfaces (denying the parallel postulate) were byond Euclid. Notably, Elements didn't have imaginary or transcendental numbers. Euclid made no attempt to unify line lengths and arc lengths, and had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers.

paulddraper 2y ago
> Elements didn't have imaginary or transcendental numbers Elements has pi.* *It proves the ratio of a circle's area to the square of its diameter is constant.
- gowld 2y ago
  But doesn't say how large that constant value is.
Someone 2y ago
> Euclid […] had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers Did he even know those gaps existed? Euclid lived around 300BC. The problem of squaring the circle had been proposed around two centuries before that (https://en.wikipedia.org/wiki/Anaxagoras#Mathematics https://en.wikipedia.org/wiki/Anaxagoras#Mathematics), but I don’t think people even considered it to be impossible by that time.

Koshkin 2y ago

Incidentally, here's one beautiful edition: https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf