Editing Ultrafinitism

{{Short description|Concept in the philosophy of mathematics}}
{{More footnotes|date=October 2015}}
In the [[philosophy of mathematics]], '''ultrafinitism''' (also known as '''ultraintuitionism''',<ref name="LCC">International Workshop on Logic and Computational Complexity, ''Logic and Computational Complexity'', Springer, 1995, p. 31.</ref> '''strict formalism''',<ref name="Iwan">St. Iwan (2000), "[https://doi.org/10.1023%2FA%3A1005651027553 On the Untenability of Nelson's Predicativism]", ''[[Erkenntnis]]'' '''53'''(1–2), pp. 147–154.</ref> '''strict finitism''',<ref name="Iwan"/> '''actualism''',<ref name="LCC"/> '''predicativism''',<ref name="Iwan"/><ref>Not to be confused with Russell's [[predicativism]].</ref> and '''strong finitism''')<ref name="Iwan"/> is a form of [[finitism]] and [[intuitionism]]. There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections to [[total function|totality]] of number theoretic functions like [[exponentiation]] over [[natural number]]s.

==Main ideas==
Like other [[finitism|finitists]], ultrafinitists deny the existence of the [[infinite set]] <math>\N</math> of [[natural numbers]], on the basis that it can never be completed (i.e., there is a largest natural number).

In addition, some ultrafinitists are concerned with acceptance of objects in mathematics that no one can construct in practice because of physical restrictions in constructing large finite mathematical objects.
Thus some ultrafinitists will deny or refrain from accepting the existence of large numbers, for example, the [[floor function|floor]] of the first [[Skewes's number]], which is a huge number defined using the [[exponential function]] as exp(exp(exp(79))), or
: <math> e^{e^{e^{79}}}. </math>
The reason is that nobody has yet calculated what [[natural number]] is the [[Floor and ceiling functions|floor]] of this [[real number]], and it may not even be physically possible to do so. Similarly, <math>2\uparrow\uparrow\uparrow 6</math> (in [[Knuth's up-arrow notation]]) would be considered only a formal expression that does not correspond to a natural number.  The brand of ultrafinitism concerned with physical realizability of mathematics is often called [[actualism]].

[[Edward Nelson]] criticized the classical conception of natural numbers because of the circularity of its definition. In classical mathematics the natural numbers are defined as 0 and numbers obtained by the iterative applications of the [[successor function]] to 0. But the concept of natural number is already assumed for the iteration. In other words, to obtain a number like <math>2\uparrow\uparrow\uparrow 6</math> one needs to perform the successor function iteratively (in fact, exactly <math>2\uparrow\uparrow\uparrow 6</math> times) to 0.

Some versions of ultrafinitism are forms of [[constructivism (mathematics)|constructivism]], but most constructivists view the philosophy as unworkably extreme.  The logical foundation of ultrafinitism is unclear; in his comprehensive survey ''Constructivism in Mathematics'' (1988), the constructive logician [[A. S. Troelstra]] dismissed it by saying "no satisfactory development exists at present." This was not so much a philosophical objection as it was an admission that, in a rigorous work of [[mathematical logic]], there was simply nothing precise enough to include.

== People associated with ultrafinitism ==
Serious work on ultrafinitism was led, from 1959 until his death in 2016, by [[Alexander Esenin-Volpin]], who in 1961 sketched a program for proving the consistency of [[Zermelo–Fraenkel set theory]] in ultrafinite mathematics.  Other mathematicians who have worked in the topic include [[Doron Zeilberger]], [[Edward Nelson]], [[Rohit Jivanlal Parikh]], and [[Jean Paul Van Bendegem]]. The philosophy is also sometimes associated with the beliefs of [[Ludwig Wittgenstein]], [[Robin Gandy]], [[Petr Vopěnka]], and [[Johannes Hjelmslev]].

[[Shaughan Lavine]] has developed a form of set-theoretical ultrafinitism that is consistent with classical mathematics.<ref name="stanford1">{{cite web|url=http://plato.stanford.edu/entries/philosophy-mathematics/ |title=Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) |publisher=Plato.stanford.edu |access-date=2015-10-07}}</ref>
Lavine has shown that the basic principles of arithmetic such as "there is no largest natural number" can be upheld, as Lavine allows for the inclusion of "indefinitely large" numbers.<ref name="stanford1"/>

== Computational complexity theory based restrictions ==
Other considerations of the possibility of avoiding unwieldy large numbers can be based on [[computational complexity theory]], as in [[András Kornai]]'s work on explicit finitism (which does not deny the existence of large numbers)<ref>[https://archive.today/20120713221118/http://kornai.com/Drafts/fathom_3.html "Relation to foundations"]</ref> and [[Vladimir Sazonov]]'s notion of [[feasible number|feasible numbers]].<!-- seems unclear whether it is really *his* concept, someone can read attached to learn more. https://link.springer.com/chapter/10.1007/3-540-60178-3_78 -->

There has also been considerable formal development on versions of ultrafinitism that are based on complexity theory, like [[Samuel Buss]]'s [[bounded arithmetic]] theories, which capture mathematics associated with various complexity classes like [[P (complexity)|P]] and [[PSPACE]]. Buss's work can be considered the continuation of [[Edward Nelson]]'s work on [[predicative arithmetic]] as bounded arithmetic theories like S12 are interpretable in [[Raphael Robinson]]'s theory [[Robinson arithmetic|Q]] and therefore are predicative in [[Edward Nelson|Nelson]]'s sense.  The power of these theories for developing mathematics is studied in bounded reverse mathematics as can be found in the works of [[Stephen A. Cook]] and [[Phuong The Nguyen]]. However these are not philosophies of mathematics but rather the study of restricted forms of reasoning similar to [[reverse mathematics]].

== See also ==
* [[Transcomputational problem]]
* [[Internal Set Theory]] — An enrichment of ZFC which has theorem's such as "there exists a largest standard natural number". Developed by ultrafinitist [[Edward Nelson]].

==Notes==
{{Reflist}}

==References==
*{{citation|mr=0147389 
|last=Ésénine-Volpine|first=A. S.|author-link=Alexander Esenin-Volpin
|chapter=Le programme ultra-intuitionniste des fondements des mathématiques|year= 1961 |title=Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959) |pages= 201–223|publisher= Pergamon|place= Oxford}} Reviewed by {{citation|title=Review of Le Programme Ultra-Intuitionniste des Fondements des Mathematiques by A. S. Ésénine-Volpine
|first1= G. |last1=Kreisel |first2=A.|last2= Ehrenfeucht
|journal=The Journal of Symbolic Logic|volume= 32|issue= 4 |year=1967|page= 517
|publisher=Association for Symbolic Logic
|doi= 10.2307/2270182
|jstor=2270182}}
* Lavine, S., 1994. [https://books.google.com/books?id=GvGqRYifGpMC&q=%22strict+finitism%22 Understanding the Infinite], Cambridge, MA: Harvard University Press.

==External links==
*[https://doi.org/10.1023%2FA%3A1024451401255  Explicit finitism] by [[András Kornai]]
*[https://doi.org/10.1007%2F3-540-60178-3_78 On feasible numbers by Vladimir Sazonov]
*[http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf "Real" Analysis Is A Degenerate Case Of Discrete Analysis]  by [[Doron Zeilberger]]
*[https://mathoverflow.net/q/44208 Discussion on formal foundations] on [[MathOverflow]]
*[https://web.archive.org/web/20060209210015/http://staff.science.uva.nl/~anne/hhhist.pdf History of constructivism in the 20th century] by [[A. S. Troelstra]]
*[http://www.math.princeton.edu/~nelson/books/pa.pdf Predicative Arithmetic] by [[Edward Nelson]]
*[http://www.cs.toronto.edu/~sacook/homepage/book/ Logical Foundations of Proof Complexity] by [[Stephen A. Cook]] and [[Phuong The Nguyen]]
*[http://www.cs.toronto.edu/~pnguyen/studies/thesis.pdf Bounded Reverse Mathematics] by [[Phuong The Nguyen]]
*[http://www.charlespetzold.com/blog/2008/05/Reading-Brian-Rotmans-Ad-Infinitum.html Reading Brian Rotman’s “Ad Infinitum…”] by [[Charles Petzold]]
*[http://plato.stanford.edu/entries/computational-complexity/ Computational Complexity Theory]

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[[Category:Constructivism (mathematics)]]
[[Category:Philosophy of mathematics]]
[[Category:Infinity]]
[[Category:Theories of deduction]]