Ultrafinitism

In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism is a form of finitism. There are various philosophies of mathematics which are called ultrafinitism. One of major identifying properties common between most of these philosophies is their denial of totality of number theoretic functions like exponentiation over natural numbers.

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Main ideas

Like other strict finitists, ultrafinitists deny the existence of the infinite set N of natural numbers, on the grounds that it can never be completed.

In addition, some ultrafinitists are concerned with our own physical restrictions in constructing (even finite) mathematical objects. Thus some ultrafinitists will deny the existence of, for example, the floor of the first Skewes' number, which is a huge number defined using the exponential function as exp(exp(exp(79))), or

e^{e^{e^{79}}}\mbox{.} \!

The reason is that nobody has yet calculated what natural number is the floor of this real number, and it may not even be physically possible to do so. Similarly, 2\uparrow\uparrow\uparrow 6 is considered only a formal expression which does not correspond to a natural number. The ultrafinitism concerned with physical realizability of mathematics is often called actualism.

Edward Nelson criticizes 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 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 of constructivism, but even constructivists generally 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 has been led, since 1959, by Alexander Esenin-Volpin. Other mathematicians who have worked in the topic include Doron Zeilberger, Edward Nelson, and Rohit Jivanlal Parikh. The philosophy is also sometimes associated with the views of Ludwig Wittgenstein, Robin Gandy and J. Hjelmslev.

Shaughan Lavine has developed a form of set-theoretical ultra-finitism that is consistent with classical mathematics.[1]

Complexity theory based restrictions

Other considerations of the possibility of avoiding unwieldily large numbers can be based on computational complexity theory, as in Andras Kornai's work on explicit finitism (which does not deny the existence of large numbers[2]) and Vladimir Sazonov's notion of feasible number.

There has been considerable formal development on complexity based views like Samuel Buss's Bounded Arithmetic theories which capture mathematics associated with various complexity classes like P and PSPACE. 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 researches are not philosophies of mathematics but rather the study of restricted forms of reasoning similar to Reverse Mathematics.

Notes

  1. ^ http://plato.stanford.edu/entries/philosophy-mathematics/
  2. ^ http://kornai.com/Drafts/fathom_3.html

References

External links