|Law of the Excluded Middle: an assertion is either true or false. "There is no third possibility."See also bivalence, anti-realism, multivalued logic.|
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|Arend Heyting Ein Streitgespräch 1956 in Kursbuch 8 Mathematik 1967
The sentence of the excluded middle/VsIntuitionism: one does not accuse the intuitionist that he accepts too little, as the representative of classical mathematics thinks, he rather accepts too much. E.g. Is the principle of the excluded middle as evident to most people as is the one of complete induction? Why does he reject the one and agrees on the other?
Intuitionism: In fact, intuitionist claims must appear dogmatic to those who regard them as assertions about facts, but they are not meant like this.
They consist of mental constructions. Mathematical ideas belong to my highly private world of thought. E.g. "I've added 2 and 3 and then 4 and i and have determined that this leads to the same result".
This does not convey any knowledge about the external world, but about my thoughts. One must distinguish between the mere practice of mathematics and its assessment. The value always depends on our philosophical ideas.
If science really tends to formalize language, then intuitionist mathematics does not belong to science in the sense of the word. Rather, it is a phenomenon of life, a natural activity of the human being. The meta-mathematical considerations may be useful; they cannot be integrated into intuitionist mathematics.
The mathematics, from the intuitionist standpoint, is the study of certain functions of the human mind.