Theory of Immensity
Saturday, 5. May 2007, 11:02:00
Primitive people who couldn't count beyond two or three, called any greater number “many”. Even though modern first-grade pupils operate numbers up to a hundred, and an adult understands what a billion is, human's ability for direct perception of quantities hasn't improved in a qualitative sense. We have ten fingers on our hands, fifty matches in a box, a hundred centimeters in a meter (both units are comparable to the sizes of body parts and therefore are available for direct perception). Starting with about a thousand, it becomes problematic to compare a number to something familiar. Your screen probably has several millions of pixels, each of which an acute-visioned person can discern. This seems like the biggest order for a number that can be directly matched against the environment. Because even secondary school education allows one to operate on numbers of greater orders, our perception lags behind the abilities of abstract thinking. Values that are orders of magnitude greater than the perception threshold feel immense.
Even though 1 000 000 000 < 1 000 000 001, the increase of one doesn't “make a difference”, and both numbers feel equally immense. However, both of them are more immense than a million. Basing on these intuitive assessments, one can say that the immensity of a number is a measure of psychological perception of its magnitude. In a similar manner to the other sensations (brightness, volume, smell or taste intensity), immensity is measured on a non-linear scale where every next equal interval corresponds to a bigger difference in input. The scale should also have a saturation threshold after which no increase of input can increase the psychological evaluation.
It would be naïve to assume that the sensation of immensity, just like vision or hearing, follows the logarithmic Weber–Fechner law. However, 10103 is hardly as much more immense than 10100 as a million is more immense than a thousand. To make as significant a leap from 10100 as the leap between a thousand and a million, one would need a value like 10200. After that, the leaps become even more drastic: to give a “worthy increase” to 1010100, it takes something like 101010100. It seems that, with even increase of immensity, the numbers have to grow faster than any arithmetic function comprehensible to the subject.
I suppose that the feeling of immensity is related to description of a method to obtain a given value from numbers below the perception threshold. For example, a billion is a thousand of thousands of thousands, 264 is 2 doubled 64 times. The more qualitatively different steps it takes to reach the goal, the more immense is the number. For example, squaring a a number is a step, and squaring a number an immense number of times is two steps (“square a number” and “repeat the last step many times”). In this regard, the Graham's number is particularly interesting: it is so big that it requires a special notation to transcribe. To reach from non-immense numbers to the Graham's number g64, it takes five qualitative transitions. The immensity of this number seems to approach the saturation threshold, and even though one could continue along the lines of gg64, it's practically impossible to increase the sense of immensity compared to the Graham's number. However, the range of perceived immensities probably depends on the subject's mathematical grounding. A fourth grade pupil's saturation threshold is hardly above two or three steps, while Ronald Graham can probably appreciate numbers even more immense than g64.
See also:
Please assess how immense these numbers are. It's your psychological sensation that is important, not the encyclopedic knowledge of the magnitudes. The left end of the immensity scale corresponds to “modest” numbers like a million, while the right end is for numbers so inconceivably big that you cannot imagine numbers that feel significantly bigger.
The number of atoms in the Universe (1…6)
The number of possible chess games without repeating positions (1…6)
The number of cells in all living organisms on Earth (1…6)
The number of people ever born (1…6)
The number of possible texts the length of “War and Peace” (1…6)
The number of seconds having elapsed since the Big Bang (1…6)
Please don't vote if you haven't read all of the above.
По-русски: Теория громадности
Even though 1 000 000 000 < 1 000 000 001, the increase of one doesn't “make a difference”, and both numbers feel equally immense. However, both of them are more immense than a million. Basing on these intuitive assessments, one can say that the immensity of a number is a measure of psychological perception of its magnitude. In a similar manner to the other sensations (brightness, volume, smell or taste intensity), immensity is measured on a non-linear scale where every next equal interval corresponds to a bigger difference in input. The scale should also have a saturation threshold after which no increase of input can increase the psychological evaluation.
It would be naïve to assume that the sensation of immensity, just like vision or hearing, follows the logarithmic Weber–Fechner law. However, 10103 is hardly as much more immense than 10100 as a million is more immense than a thousand. To make as significant a leap from 10100 as the leap between a thousand and a million, one would need a value like 10200. After that, the leaps become even more drastic: to give a “worthy increase” to 1010100, it takes something like 101010100. It seems that, with even increase of immensity, the numbers have to grow faster than any arithmetic function comprehensible to the subject.
I suppose that the feeling of immensity is related to description of a method to obtain a given value from numbers below the perception threshold. For example, a billion is a thousand of thousands of thousands, 264 is 2 doubled 64 times. The more qualitatively different steps it takes to reach the goal, the more immense is the number. For example, squaring a a number is a step, and squaring a number an immense number of times is two steps (“square a number” and “repeat the last step many times”). In this regard, the Graham's number is particularly interesting: it is so big that it requires a special notation to transcribe. To reach from non-immense numbers to the Graham's number g64, it takes five qualitative transitions. The immensity of this number seems to approach the saturation threshold, and even though one could continue along the lines of gg64, it's practically impossible to increase the sense of immensity compared to the Graham's number. However, the range of perceived immensities probably depends on the subject's mathematical grounding. A fourth grade pupil's saturation threshold is hardly above two or three steps, while Ronald Graham can probably appreciate numbers even more immense than g64.
See also:
- S. Kozlovsky. The Biggest Number in the World (in Russian).
- E. Yudkovsky. Systematic Mistakes in Reasoning Potentially Affecting the Evaluation of Global Risks (in Russian), see “8. Scope neglect”.
Please assess how immense these numbers are. It's your psychological sensation that is important, not the encyclopedic knowledge of the magnitudes. The left end of the immensity scale corresponds to “modest” numbers like a million, while the right end is for numbers so inconceivably big that you cannot imagine numbers that feel significantly bigger.
The number of atoms in the Universe (1…6)
The number of possible chess games without repeating positions (1…6)
The number of cells in all living organisms on Earth (1…6)
The number of people ever born (1…6)
The number of possible texts the length of “War and Peace” (1…6)
The number of seconds having elapsed since the Big Bang (1…6)
Please don't vote if you haven't read all of the above.
По-русски: Теория громадности
