More About Proofs

Every one knows that mathematics is about proving theorems. Mathematics is the only deductive science and both the study and development of mathematics revolve around deducing one thing from another. Several misconceptions involving the notion of proof are quite common:

1. A proof is a deduction of facts from other, simpler facts. After all, Euclid built the body of Geometry on the foundation of 5 simple facts.

   In reality, many math facts have been proven to be equivalent. Equivalency means that each of the two facts can be derived from the other. In what sense then one is simpler? Proof may be about seeing something in a different light or finding unexpected links.

2. This is how one develops mathematics - picking simple facts and deriving from them something new.

   This is actually never the case. Most often a mathematician senses that something is true, believes in it and a proof is a way of communicating the idea by giving it a sound foundation and, perhaps, explaining its origins. A proof transforms an idea into a fact. Quite often, attempts to prove an idea lead to its refutal.

3. Whatever is the right definition of the proof (most of us feel it in the guts when something is doubtful while something else is certain, i.e. proven), every one agrees with the definition. The definition is immutable.

   This is not true for several reasons. For example, there always were mathematicians who feel uncomfortable with the notion of infinity. Using the Axiom of Choice Tarski and Banach have shown that it's possible to split a tennis ball into a few parts that could be combined again only to produce a ball of the size of our Earth. Many feel that the axiom is not a reliable tool. However, most mathematicians would not hesitate to use it.

   The notion of proof underwent very dramatic changes in the 19th century when, for many reasons, mathematicians began questioning their intuition. Euclid's Elements define a straight line as a breadthless length. In 1945, in a Russian math olympiad, an 8th grade boy who did not even attempt to solve but one problem received a first prize for a remark he submitted with an unfinished proof of that problem:

   I spent much time trying to prove that a straight line can't intersect three sides of a triangle in their interior points but failed for, to my consternation, I realized that I have no notion of what a straight line is.

   In the 19th century mathematicians who questioned the 2300 year old intuition about straight lines discovered non-Euclidean geometries that, in the 20th century, were incorporated by Einstein into his General Theory of Relativity.

   If intuition permeates any mathematical activity, how is it possible to build a theory that prides itself in its rigor on top of something rather elusive and individual? A historic example will shed some light on how this happens.

• 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