HPS391H1 Chapter Notes - Chapter 6: Algebraic Equation, Real Number, Georg Cantor
29 Apr 2016
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Newton, lebniz and euler are seen as the (cid:858)fathers(cid:859) of analysis in the late seventeenth and eighteenth centuries. Though their work was seen as revolutionary, and to large extent logically rational, there was a very evident problem in the underpinnings of their (cid:272)al(cid:272)ulus, (cid:449)hi(cid:272)h la(cid:455) i(cid:374) the use of (cid:858)i(cid:374)fi(cid:374)itel(cid:455) s(cid:373)all/la(cid:396)ge(cid:859) (cid:395)ua(cid:374)tities pe(cid:396)tai(cid:374)i(cid:374)g to li(cid:373)its. Ne(cid:449)to(cid:374) (cid:449)as the fi(cid:396)st to gi(cid:448)e his defi(cid:374)itio(cid:374). His theo(cid:396)(cid:455) of (cid:858)flu(cid:454)io(cid:374)s(cid:859) (cid:449)as (cid:272)e(cid:374)te(cid:396)ed o(cid:374) observing the ratio between two quantities as they both simultaneously approached zero (in modern terminology, observing the limit of their ratio), which he called the (cid:858)ulti(cid:373)ate (cid:396)atio(cid:859). Ho(cid:449)e(cid:448)e(cid:396), if the (cid:396)atio app(cid:396)oa(cid:272)hes (cid:894) ), what meaning does this. Lei(cid:271)(cid:374)iz (cid:271)ased his defi(cid:374)itio(cid:374) of li(cid:373)its o(cid:374) (cid:858)i(cid:374)fi(cid:374)itel(cid:455) s(cid:373)all(cid:859) (cid:395)ua(cid:374)tities (cid:894)app(cid:396)oa(cid:272)hi(cid:374)g ze(cid:396)o, definition have? but cannot be decreased further; the next closest thing to zero). However this definition was clearly ambiguous and not mathematically well-defined.
