The Banach–Tarski paradox is impossible in two dimensions. Therefore it demonstrates simply the limitations and inconsistency of different forms of human reasoning.

Mathematics speaks about the forms of human reasoning whereas physics tries to speak about reality. However, it is impossible to speak about anything other than in words which are records and instances of reasoning.

Clearly, reality will remain without humans, while mathematics would disappear had mankind extinguished. However the same fate is doomed for physics, demonstrating the common nature of mathematics and physics. Physics and mathematics belong to mankind, both dealing primarily with the human visions of the world rather than the world itself.

Homo sapiens is a part of reality together with his brain and mentality. Neither science nor mathematics is possible without humans. So all ado about the ostensible abyss between science and mathematics is groundless. Inconsistency of ideas may be thought of as a gain rather then a loss of man.

Apparently, there is nothing paradoxical about the Banach–Tarski paradox. It is simply another proof of the flexibility and enterprise of human’s mind.

S. Kutateladze

September 25, 2009

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© Kutateladze S. S. 2009