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On Frege seeing what is in front of his nose

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Take a mathematician of Frege’s generation, accustomed to writing the likes (1) (2) If , then or , — and fancier things, of course! Whatever unclear thoughts about ‘variables’ people may or may not have had once upon a time, they have surely been dispelled well before the 1870s, if not by Balzano’s 1817 Rein analytischer Beweis (though perhaps that was not widely enough read?), at least by Cauchy’s 1821 Cours d’analyse which everyone serious will have read. Both Bolzano and Cauchy take claims like (1) and (2) to be true just if each instance is true, plain and simple, and clearly gloss various claims written with variables as claims holding ‘for any value of x’. Maybe it is worth noting, then, that the mathematicians of the day, at least when on their best behaviour, could be very decently clear about this! But then it seems to be only the tiniest of steps to say outright that an ideal notation for such claims might have the form. . .

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News source: Logic Matters

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