Ultrafinitism

Ultrafinitism (also called ultraintuitionism ) is a particularly strict interpretation of finitism in the philosophy of mathematics .

Ultrafinitists deal with the construction of mathematical objects taking into account the physical limitations of humans. Not only do they, like other followers of finitism , reject the existence of mathematical structures, the construction of which would require an infinite number of steps (as with the set of natural numbers ), but they also doubt the meaning of numbers such as (the greatest natural Number smaller than Skewes' number ). They justify this with the fact that nobody has calculated this number yet and that it may not be physically possible to calculate it. The number of elementary particles in the observable universe is estimated at , while Skewes' number is at . ${\displaystyle \mathbb {N} }$ ${\displaystyle \left\lfloor e^{e^{e^{79}}}\right\rfloor }$ ${\displaystyle 10^{100}=10^{10^{2}}}$ ${\displaystyle 10^{10^{10^{34}}}}$

Although ultrafinitism is a form of mathematical constructivism , it is viewed by the vast majority of constructivists as impractical. The constructive logician AS Troelstra formulated this in Constructivism in Mathematics ( 1988 ): “There is currently no satisfactory development” (“no satisfactory development exists at present”).

Alexander Jessenin-Wolpin had been a leading exponent of ultrafinitism since 1959 .

Web links

• Andras Kornai , Explicit finitism, International Journal of Theoretical Physics 42, 301-307 (2003), doi: 10.1023 / A: 1024451401255
• Vladimir Sazonov , On feasible numbers, in: International Workshop on Logic and Computational Complexity (LCC), pp. 30–51, doi: 10.1007 / 3-540-60178-3_78 ( [1] )
• "Real" Analysis Is A Degenerate Case Of Discrete Analysis An introduction to an ultrafinitist perspective, English, PDF; 90 kB by Doron Zeilberger
• Discussion on formal foundations , by MathOverflow
• History of constructivism in the 20th century , by AS Troelstra
• Predicative Arithmetic English, by Edward Nelson
• Logical Foundations of Proof Complexity , by Stephen A. Cook and Phuong The Nguyen
• Bounded Reverse Mathematics English, by Phuong The Nguyen
• "Already the potential infinity is metaphysics" English, criticism of the book Ad-Infinitum by Brian Rotman by Charles Petzold