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What is the point of formalization in mathematics and how does it relate to axiomatization?

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Formalization is often presented (1) as a further stage after axiomatization. (It is often said that euclidian geometry was already axiomatized, but not yet formalized ( or not enough), and that the complete formalization of geometry was only achieved much later, by Tarsky for example) (2) it is is also presented as the full completion of axiomatization.

Reference : " if geometry is to be deductive, the deduction must everywhere be independant of the meaning of geometrical concepts , just as it must be independant of the diagrams; only the relations specified in the propositions and definitions employed may legitimately be taken into account" Pasch ( quoted by Wilder, Introduction To The Foundations Of Mathematics, Part I The axiomatic method, I, §1 Evolution of the method. )

My question is : why is this further stage needed? why formalizing mathematics? ( what is the interest in ordinary mathematical practice? why should a fully axiomatized mathematical theory be also formalized?)

Remark.- My question is not equivalent to " is the formalist view of mathematics correct?" I'm looking for an answer mathematicians could agree on , whatever might be their own commitment to a particular philosophy of mathematics?

logic axioms

share|cite|improve this question

edited 2 days ago

Ray LittleRock

asked 2 days ago

Ray LittleRockRay LittleRock

789

$endgroup$

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  On Hilbert : not correct. Hilbert achieved in 1899 a more rigorous i.e. "complete" (with respect to Euclid's original one) axiomatization of Geometry. The tretment was not formalized.
  $endgroup$
  – Mauro ALLEGRANZA
  2 days ago
  

  
  
  
  


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  1
  

  
  
  
  
  
  

  
  $begingroup$
  A "more formal" system is due to Tarski. See : Alfred Tarski, What is Elementary Geometry ? (1959) as well as Alfred Tarski & Steven Givant, arski's System of Geometry (1999).
  $endgroup$
  – Mauro ALLEGRANZA
  2 days ago
  

  
  
  
  


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  $begingroup$
  Formalization allows us to study the meta-mathematical properties of mathematical theories : completeness, consistency, decidability. The aim of formalization is to represent the math theory as a mathematical object itself, and thus to make precise questions about its mathematical properties. See Proof Theory and Model Theory.
  $endgroup$
  – Mauro ALLEGRANZA
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mauro Allegranza Very helpfull!
  $endgroup$
  – Ray LittleRock
  2 days ago
  

  
  
  
  


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  1
  

  
  
  
  
  
  

  
  $begingroup$
  See en.wikipedia.org/wiki/Entscheidungsproblem
  $endgroup$
  – John Douma
  2 days ago
  

  
  
  
  

 | 
show 1 more comment

0

$begingroup$

Formalization is often presented (1) as a further stage after axiomatization. (It is often said that euclidian geometry was already axiomatized, but not yet formalized ( or not enough), and that the complete formalization of geometry was only achieved much later, by Tarsky for example) (2) it is is also presented as the full completion of axiomatization.

Reference : " if geometry is to be deductive, the deduction must everywhere be independant of the meaning of geometrical concepts , just as it must be independant of the diagrams; only the relations specified in the propositions and definitions employed may legitimately be taken into account" Pasch ( quoted by Wilder, Introduction To The Foundations Of Mathematics, Part I The axiomatic method, I, §1 Evolution of the method. )

My question is : why is this further stage needed? why formalizing mathematics? ( what is the interest in ordinary mathematical practice? why should a fully axiomatized mathematical theory be also formalized?)

Remark.- My question is not equivalent to " is the formalist view of mathematics correct?" I'm looking for an answer mathematicians could agree on , whatever might be their own commitment to a particular philosophy of mathematics?

logic axioms

share|cite|improve this question

edited 2 days ago

Ray LittleRock

asked 2 days ago

Ray LittleRockRay LittleRock

789

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  On Hilbert : not correct. Hilbert achieved in 1899 a more rigorous i.e. "complete" (with respect to Euclid's original one) axiomatization of Geometry. The tretment was not formalized.
  $endgroup$
  – Mauro ALLEGRANZA
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  A "more formal" system is due to Tarski. See : Alfred Tarski, What is Elementary Geometry ? (1959) as well as Alfred Tarski & Steven Givant, arski's System of Geometry (1999).
  $endgroup$
  – Mauro ALLEGRANZA
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Formalization allows us to study the meta-mathematical properties of mathematical theories : completeness, consistency, decidability. The aim of formalization is to represent the math theory as a mathematical object itself, and thus to make precise questions about its mathematical properties. See Proof Theory and Model Theory.
  $endgroup$
  – Mauro ALLEGRANZA
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mauro Allegranza Very helpfull!
  $endgroup$
  – Ray LittleRock
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  See en.wikipedia.org/wiki/Entscheidungsproblem
  $endgroup$
  – John Douma
  2 days ago
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Formalization is often presented (1) as a further stage after axiomatization. (It is often said that euclidian geometry was already axiomatized, but not yet formalized ( or not enough), and that the complete formalization of geometry was only achieved much later, by Tarsky for example) (2) it is is also presented as the full completion of axiomatization.

Reference : " if geometry is to be deductive, the deduction must everywhere be independant of the meaning of geometrical concepts , just as it must be independant of the diagrams; only the relations specified in the propositions and definitions employed may legitimately be taken into account" Pasch ( quoted by Wilder, Introduction To The Foundations Of Mathematics, Part I The axiomatic method, I, §1 Evolution of the method. )

My question is : why is this further stage needed? why formalizing mathematics? ( what is the interest in ordinary mathematical practice? why should a fully axiomatized mathematical theory be also formalized?)

Remark.- My question is not equivalent to " is the formalist view of mathematics correct?" I'm looking for an answer mathematicians could agree on , whatever might be their own commitment to a particular philosophy of mathematics?

logic axioms

share|cite|improve this question

edited 2 days ago

Ray LittleRock

asked 2 days ago

Ray LittleRockRay LittleRock

789

$endgroup$

share|cite|improve this question

edited 2 days ago

Ray LittleRock

asked 2 days ago

Ray LittleRockRay LittleRock

789

share|cite|improve this question

edited 2 days ago

Ray LittleRock

asked 2 days ago

Ray LittleRockRay LittleRock

789

asked 2 days ago

Ray LittleRockRay LittleRock

789

asked 2 days ago

Ray LittleRockRay LittleRock

789