QUESTION 1

Date sent: Mon, 25 Sep 2000
From: Matthias Kissel

Sehr geehrter Herr von Glasersfeld,

könnten Sie mir bitte wenigstens ein möglichst einfaches Beispiel geben für den Beweis eines mathematischen Theorems unter Verwendung imaginärer logischer Werte? Ich suche schon lange danach,bisher leider vergeblich.

Danke!

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Dear Mr. Kissel,

I answer your question in English because I am sure you won't have difficulties reading it.

When you say: "imaginary logical values", I take it to mean "imaginary numbers" - I don't know what else it should mean.

Let me state emphatically: I am not a mathematician - but I have J.R.Newman's "The World of Mathematics", a wonderful book that has many times saved my life.

In it, I quickly found an answer to your question: The expression "(a square + b square) multiplied by (c square plus d square)" can transformed into "(a+bi)(a-bi) (c+di)(c-bi), where "i" = root of -1; if you then carry out the indicated operations, you get the equation: "(ac - bd)squared + (ad + bc)squared". Which is considered "an interesting and easily verifiable theorem on numbers". This was easy enough for me to understand, so it's easy! Any mathematician will give you more sophisticated examples. 

Best wishes, Ernst von Glasersfeld