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\begin{document}

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\section*{finite mathematics}

\hypertarget{finite_mathematics}{}\section*{{Finite mathematics}}\label{finite_mathematics}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{finitism}{Finitism}\dotfill \pageref*{finitism} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Finite mathematics} is the [[mathematics]] of [[finite set]]s. The term is sometimes used more broadly of [[discrete mathematics]].

We may say that finite mathematics is mathematics done [[internalization|internal]] to the category [[FinSet]] of finite sets or the mathematics of $\Fin\Set$ itself. The latter (but not the former) includes the basic arithmetic of [[natural number]]s, since these are the [[cardinal numbers|cardinalities]] of finite sets; we can go as far as rational numbers this way, but not [[real number]]s. This also includes a great deal of combinatorics, basic algebra, and elementary logic, although not many advanced topics.

\hypertarget{finitism}{}\subsection*{{Finitism}}\label{finitism}

In the [[foundations]] of mathematics and in [[philosophy of mathematics]], \textbf{finitism} is the philosophy that one should do only finite mathematics. In a weak sense, one should not assume the [[axiom of infinity]]; in a strong sense, one should even deny it by an axiom of finiteness. This make it impossible to do analysis as we normally understand it.

Finitism (in the weak sense of not accepting an axiom of infinity) is essentially the mathematics that can be done internal to an arbitrary [[boolean topos]] (at least if one is not also being [[predicative mathematics|predicative]] or [[constructive mathematics|constructive]]). For [[constructive mathematics]] as usually practised, one goes beyond finitism by positing a [[natural numbers object]].

Although often considered an extreme form of constructivism, finitism in the strong sense (actually denying the axiom of infinity) can make [[excluded middle]] and even the [[axiom of choice]] constructively acceptable (and similarly make [[power sets]] predicatively acceptable). This is because even constructivists agree that these are true in $\Fin\Set$; it's the extension of them to [[infinite set]]s that the first constructivists objected to.

\textbf{Ultrafinitism} is an even more extreme form of finitism, in which one doubts the existence even of very large numbers, numbers which in some sense it is not physically possible to write down. The theory of ultrafinite mathematics is most well developed by [[Edward Nelson]], although foundational systems such as [[soft linear logic]] can also be argued to have an ultrafinitist flavor.

For the opinionated espousal of finitism (and much else), one can hardly do better than the \href{http://www.math.rutgers.edu/~zeilberg/OPINIONS.html}{Opinions} of [[Doron Zeilberger]].

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Edward Nelson]], \emph{Warning Signs of a Possible Collapse of Contemporary Mathematics} (\href{https://web.math.princeton.edu/~nelson/papers/warn.pdf}{pdf})

\end{itemize}
[[!redirects finite mathematics]]

[[!redirects finitist mathematics]] [[!redirects finitism]] [[!redirects ultrafinitism]]



\end{document}