If PIC were a mathematical question (or problem)—if it were finitistically restricted it would be algorithmically decidable, which it is not [ RFM V, §21), (LFM 31 32, 111, 170), (WVC 102 03)].
sig=lfm+0.2*randn(length(lfm),1)'; % add gaussian noise.
This giddiness and pleasant feeling of paradox, says Wittgenstein (LFM 16), "may be the chief reason [set theory] was invented".
This fact is brought out by the fact "that the technique of learning ℵ0 numerals is different from the technique of learning 100,000 numerals" (LFM 31).
Platonism is either "a mere truism" (LFM 239), Wittgenstein says, or it is a 'picture' consisting of "an infinity of shadowy worlds" (LFM 145), which, as such, lacks 'utility' (cf. PI §254) because it explains nothing and it misleads at every turn.
By means of Wittgenstein's "immensely important" 'investigation' (LFM 103), we will find, Wittgenstein expects, that set theory is uninteresting (e.g., that the non-enumerability of "the reals" is uninteresting and useless) and that our entire interest in it lies in the 'charm' of the mistaken prose interpretation of its proofs (LFM 16).
Akabori et al. studied the surface relaxation behaviors in PS films with different thicknesses by lateral force microscopy (LFM) [17].
For example, Akabori et al. demonstrated how the surface relaxation behavior depended on the thickness of PS films using lateral force microscopy (LFM) [10].
The conversion efficiency, i.e., (frac{mathrm { #Order}}{mathrm { Rcomds.}}), of our methods is 3.99%, which is better than LFM: 3.20%.
Largely a product of his anti-foundationalism and his criticism of the extension-intension conflation, Wittgenstein's later critique of set theory is highly consonant with his intermediate critique [ PR §§109, 168), (PG 334, 369, 469), (LFM 172, 224, 229), and (RFM III, §43, 46, 85, 90; VII, §16)].
"[I]t is essential to mathematics that its signs are also employed in mufti," Wittgenstein states, for "[i]t is the use outside mathematics, and so the meaning ['Bedeutung'] of the signs, that makes the sign-game into mathematics" (i.e., a mathematical "language-game") [ RFM V, §2, 1942), (LFM 140 141, 169 70)].
