Ultrafinitism

In thephilosophy of mathematics,ultrafinitism(also known asultraintuitionism,^[1]strict formalism,^[2]strict finitism,^[2]actualism,^[1]predicativism,^[2]^[3]andstrong finitism)^[2]is a form offinitismandintuitionism.There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections tototalityof number theoretic functions likeexponentiationovernatural numbers.

Main ideas

Like otherfinitists,ultrafinitists deny the existence of theinfinite set $\mathbb {N}$ ofnatural 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, thefloorof the firstSkewes's number,which is a huge number defined using theexponential functionas exp(exp(exp(79))), or

e^{e^{e^{79}}}.

The reason is that nobody has yet calculated whatnatural numberis thefloorof thisreal number,and it may not even be physically possible to do so. Similarly, $2\uparrow \uparrow \uparrow 6$ (inKnuth'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 calledactualism.

Edward Nelsoncriticized 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 thesuccessor functionto 0. But the concept of natural number is already assumed for the iteration. In other words, to obtain a number like $2\uparrow \uparrow \uparrow 6$ one needs to perform the successor function iteratively (in fact, exactly $2\uparrow \uparrow \uparrow 6$ times) to 0.

Some versions of ultrafinitism are forms ofconstructivism,but most constructivists view the philosophy as unworkably extreme. The logical foundation of ultrafinitism is unclear; in his comprehensive surveyConstructivism in Mathematics(1988), the constructive logicianA. S. Troelstradismissed 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 ofmathematical 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, byAlexander Esenin-Volpin,who in 1961 sketched a program for proving the consistency ofZermelo–Fraenkel set theoryin ultrafinite mathematics. Other mathematicians who have worked in the topic includeDoron Zeilberger,Edward Nelson,Rohit Jivanlal Parikh,andJean Paul Van Bendegem.The philosophy is also sometimes associated with the beliefs ofLudwig Wittgenstein,Robin Gandy,Petr Vopěnka,andJohannes Hjelmslev.

Shaughan Lavinehas developed a form of set-theoretical ultrafinitism that is consistent with classical mathematics.^[4] 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.^[4]

Computational complexity theory based restrictions

Other considerations of the possibility of avoiding unwieldy large numbers can be based oncomputational complexity theory,as inAndrás Kornai's work on explicit finitism (which does not deny the existence of large numbers)^[5]andVladimir Sazonov's notion offeasible number.

There has also been considerable formal development on versions of ultrafinitism that are based on complexity theory, likeSamuel Buss'sbounded arithmetictheories, which capture mathematics associated with various complexity classes likePandPSPACE.Buss's work can be considered the continuation ofEdward Nelson's work onpredicative arithmeticas bounded arithmetic theories like S12 are interpretable inRaphael Robinson's theoryQand therefore are predicative inNelson's sense. The power of these theories for developing mathematics is studied in bounded reverse mathematics as can be found in the works ofStephen A. CookandPhuong The Nguyen.However these are not philosophies of mathematics but rather the study of restricted forms of reasoning similar toreverse mathematics.

Notes

^^a ^bInternational Workshop on Logic and Computational Complexity,Logic and Computational Complexity,Springer, 1995, p. 31.
^^a ^b ^c ^dSt. Iwan (2000), "On the Untenability of Nelson's Predicativism",Erkenntnis53(1–2), pp. 147–154.
^Not to be confused with Russell'spredicativism.
^^a ^b"Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)".Plato.stanford.edu.Retrieved2015-10-07.
^"Relation to foundations"

References

Ésénine-Volpine, A. S.(1961), "Le programme ultra-intuitionniste des fondements des mathématiques",Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959),Oxford: Pergamon, pp. 201–223,MR 0147389Reviewed byKreisel, G.; Ehrenfeucht, A. (1967), "Review of Le Programme Ultra-Intuitionniste des Fondements des Mathematiques by A. S. Ésénine-Volpine",The Journal of Symbolic Logic,32(4), Association for Symbolic Logic: 517,doi:10.2307/2270182,JSTOR 2270182
Lavine, S., 1994.Understanding the Infinite,Cambridge, MA: Harvard University Press.

External links

[LCC-1] International Workshop on Logic and Computational Complexity,Logic and Computational Complexity,Springer, 1995, p. 31.

[Iwan-2] St. Iwan (2000), "On the Untenability of Nelson's Predicativism",Erkenntnis53(1–2), pp. 147–154.

[3] Not to be confused with Russell'spredicativism.

[stanford1-4] "Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)".Plato.stanford.edu.Retrieved2015-10-07.

[5] "Relation to foundations"

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