Friedrich Waismann Suchen und Finden in der Mathematik 1938 in Kursbuch 8 Mathematik 1967
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System/Aspects/Provability/Waismann: One could also have proved the equivalences in Russell's system
~p ⇔ ~p v ~p
p v q ⇔ ~(~p v ~p ) v~(~q v ~q)
but would one have expressed with this Sheffer's discovery? Not at all! One could speak of the discovery of a new aspect.
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Sheffer stroke.
Again the question: could one look for this aspect? No. That something can be seen in this way can only be seen when it is seen.
That one aspect is possible is only seen when it is there. You can simply underline the newly discovered, so you give a new sign.
The formulas with the underlining do something different than those without underlining, they make the new structure visible.
E.g. Suppose there is a tribe of people somewhere who owns our decimal system, and calculates exactly as we do, but infinite decimal fractions remain unknown to them. People stop the division, e.g. at the 5th digit.
1/3 = 0.333333.
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The periodicity would not be noticeable to them, they would not have to think that this always goes on like this.
After the discovery of the infinite decimal fractions one "sees" the calculation differently! This is the discovery that one sees the infinite possibility of progressing into the calculation.
The emphasis on the return of the rest is the expression that he has discovered the induction. We must not forget that the division with underlining is a different type of calculation.
E.g. (5 + 3)² = 5² +2 x 5 x 3 + 3². According to one calculation this is at the same time a proof for (7 +8)²= 7² + 2 x 7 x 8 + 8², but not according to the other calculation!
We would sometimes have to underline the different digits, sometimes underline them twice.
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For example, is the x times x not the same as x²? It is a new system.
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Calculus.
