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Can mathematics be dangerous?

Mathematics deals with finite and infinite collections of objects.

Its strength and beauty is based on exact reasoning. Thus from a

mere logical point of view, mathematics is utmost fragile: A tiny little

mistake ends up with a collapse of the whole building.

Recall that mathematical induction reaches infinity within two steps:

1. the inductive anchorage,

2. the inductive move.

The following example shows how induction, if misapplied, is able to

remove any finite number of objects from the scene.

Theorem. For a given property P , and a finite set M , every element

of M has property P .

Proof. We proceed by induction on the number n of elements of

M . For n = 0, there are no elements of M , so there is nothing

to prove. Assume that the theorem holds for n-element sets, and

consider a set M = {x1 , . . . , xn+1 } with n + 1 elements. Thus by

our inductive hypothesis, the n-element subsets M1 := {x1 , . . . , xn }

and M2 := {x2 , . . . , xn+1 } meet the claim, i. e. their elements have

property P . Consequently, the elements of the union M = M1 ∪ M2

also have property P , which completes the inductive step from n to

n + 1. Whence the theorem is proved.

Remark. The theorem becomes dangerous if property P is chosen to

be “non-existent”. With this property P , the theorem yields:

Corollary. Every finite set M is empty.

What has been proved? “Parvus error in principio magnus est in

fine” - Logically, there is no difference between small and big mistakes!

Yes - logically. However, the sense to recognize mathematical beauty

is the same that prevents mathematicians from ending up in a chaos of

mistakes - it’s intuition! Just like in music: Rules are the indispensible

framework to cultivate intuition. Without intuition, mathematical

rules appear to be frightening. - In the light of intuition, they reveal

mathematical beauty.

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