Ultrafinitism is the antithesis of googology; it is the mathematical belief that the set of natural numbers is finite, and at some point numbers become large enough that they simply cease to exist or that the largest number \(+1\) gives the smallest number or the largest negative number.[1] The philosophy is an extreme form of finitism, which states that not only infinite objects (such as the transfinite ordinals) are nonexistent, but also not all finite objects exist. Ultrafinitists state that the natural numbers have an ending, because they have no physical realization and/or cannot be computed.
The primary argument for ultrafinitism is that large natural numbers are circularly defined. A natural number is defined by applying the successor function to zero a certain number of times. And here we run into a bootstrapping issue: a googolplex is S...S0 where the number of copies of S is googolplex.
Sometimes the concept of ultrafinitism may be useful in computers, for example, the largest signed integer number in 32-bit computers is 2147483647.
Episode[]
According to Scott Aaronson, Harvey Friedman, whose works such as TREE sequence and transcendental integers are well-known in googology, was attending a talk by an ultrafinitist Alexander Yessenin-Volpin:[2]
- So Friedman raised the obvious “draw the line” objection: in the sequence 21,22,…,2100, which is the first integer that Yessenin-Volpin would say doesn’t exist?
- Yessenin-Volpin asked Friedman to be more specific.
- “Okay, then. Does 21 exist?”
- Yessenin-Volpin quickly answered “yes.”
- “What about 22?”
- After a noticeable delay: “yes.”
- “23?”
- After a longer delay: “yes.”
- It soon became clear that Yessenin-Volpin would answer “yes” to every question, but would take twice as long for each one as for the one before it.
Sources[]
- ↑ Horston, Leon. Philosophy of Mathematics. Retrieved April 2013.
- ↑ Scott Aaronson, And they say complexity has no philosophical implications, The Blog of Scott Aaronson.