Intuition and Rigor

One of the torch bearers of the formalization attempts in the 19th century was the Czech analyst Bernhard Bolzano (1781-1848.) In his critique of the attempts to prove the Fundamental Theorem of Algebra, he wrote

By which he of course meant that reliance on the geometrical intuition is an unacceptable tool in deriving analytic truths. He clearly accepts the statement as true but objects to the fact of its being used offhandedly, as a self-evident truth. In the article, Bolzano proceeds to justify the statement that is variably now known as Bolzano's or the Intermediate Value Theorem.

His proof depends on the definition of continuity by Cauchy from which he derives the Sign Preserving Property of Continuous Functions. Assuming that at the left end of an interval the function is negative, he observes that it stays negative on a certain bounded set but not for the points near the second end of the interval. He continues

I am not critical of Bolzano; for it's easy to see faults from the distance of 180 years. I only want to demonstrate how painful the labors were in delivering the present day mathematics. Bolzano ends his proof with the following remark:

I have a feeling that the actual quantity is somehow related to a point on a (geometric) line. In any event, Richard Dedekind (1831-1916) might not have had the "correct concept of quantity" for in 1872 he published the first rigorous introduction into the theory of real numbers. The claimed theorem has been indeed proven on the foundation of Dedekind's Cuts.

The property claimed by the theorem is known as completeness of the set of real numbers. Depending on how deep one wants to make a presentation of mathematics, the theorem may as well be declared an axiom - the fact that requires no proof in that particular setting. However, in the footsteps of Bolzano, it would be "an intolerable offense against correct method" to refer to somebody's concept of quantity in a proof claimed to be rigorous.

References

1. The History of Mathematics, ed J.Fauvel and J.Gray, The Open University, 1987

• Perfect numbers are complex, complex numbers might be perfect
• Fundamental Theorem of Algebra: Statement and Significance
• What's in a proof?
• More about proofs
• Axiomatics
• Intuition and Rigor
• How to Prove Bolzano's Theorem
• Early attempts
• Proofs of the Fundamental Theorem of Algebra
  • Remarks on Proving The Fundamental Theorem of Algebra
    • Sketch of a Proof by Birkhoff and MacLane
    • Sketch of Second Proof (after Cauchy)
    • Details of the Cauchy's Proof
    • Note on the Extreme Value Theorem
    • Sketch of Proof by the methods of the theory of Complex Variables (after Liouville)
    • Maximum Modulus Theorem
  • A Proof of the Fundamental Theorem of Algebra: Standing on the shoulders of giants
  • Yet Another Proof of the Fundamental Theorem of Algebra
  • Fundamental Theorem of Algebra - Yet Another Proof
  • A topological proof, going in circles and counting