Frege's Approach to the Foundations of Analysis (1874–1903)

The Logical System of Frege's Grundgestze: A Rational Reconstruction

“…It follows that θ is nothing but what stands under the symbol ' − − ' in θ. 37 The rule of fusion of horizontals is embodied in Frege's notation, with the result of blurring the twofold role it plays in FGGBS, as a rule for the formation of terms and as a deductive rule. This rule is however both appealed to during the explanation of the function-names ' -ξ', ' ξ' and ' 38 In stating this rule, we could have omitted both ' − → κ2 ⇒' and '⇒ − → κ3', without any relevant consequence for its deductive strength.…”

“…3 Notice, however, that whereas the theory of natural numbers should have merely required appropriate definitions, since those should have been enough for warranting the existence of these numbers, that of real numbers should have also required an existence proof for domains of magnitudes (these numbers being identified by Frege with ratios of magnitudes), since their existence would have not been warranted by their definition ( [38]; [12], ch. 22; [37]; [39]). 4 In Frege's parlance-which we shall also adopt here-first-level functions are those whose arguments are required to be objects, while n-level functions (n = 2, 3, .…”

Abstraction and Epistemic Economy

“…105-113; [15], part III, § § II.55-245; [33]; [13], ch. 22; [30]). According to Frege, real numbers originate in measuring magnitudes, and they have to be defined as ratios of them, rather than as arithmetical items (like it happens in Cantor's and Dedekind's definitions, instead).…”

Frege's Theory of Real Numbers: A consistent Rendering

“…Useful accounts of it can be found inDummett (1991, ch. 22);Schirn (2013);Simons (1987);Shapiro & Snyder (2020).5 Anyone supporting Quine's view on the non-logicality of higher-order logic can take our granting it as made for the sake of the argument.…”