mbd_map 19: A Dedication homepage homepage forum lectures 1: A Word of Encouragement 2: Dar al-Hikma 3: Proclus' Elements 4: Reversion in the Corporeal 5: Mathematical Recursion 6: Episodic Memory 7: Mortality 7 Supplement: Classical Mortality Arguments 8: Personal Identity 9: Existential Passage 10: Precedent at Dar al-Hikma 10 Supplement: Images of Dar al-Hikma 11: Passage Types 12: A Metaphysical Grammar 13: Merger Probability 14: Ex Nihilo Probability 15: Noetic Reduction 16: Summary of Mathematical Results 17: Application to Other Species 18: Potential Benefits 19: A Dedication appendices works cited
 

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A Word of Encouragement

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Dar al-Hikma

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Proclus' Elements

4

Reversion in the Corporeal

5

Mathematical Recursion

6

Episodic Memory

7

Mortality

7s

Classical Mortality Arguments

8

Personal Identity
1   2   3   4  

9

Existential Passage
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10

Precedent at Dar al-Hikma

10s

Images of Dar al-Hikma

11

Passage Types

12

A Metaphysical Grammar

13

Merger Probability

14

Ex Nihilo Probability

15

Noetic Reduction

16

Summary of Mathematical Results

17

Application to Other Species
1   2   3   4  

18

Potential Benefits

19

A Dedication

Appendices

Works Cited



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Chapter 16
Summary of Mathematical Results


Chapters 13-15 have derived three aggregate mathematical results.  I should emphasize that these results can apply only to populations meeting the conditions assumed at the start of Chapter 13 — the assumption of population stability being especially critical.[1]
       Summarizing the results in the order of their derivation:



Question:
       What is the experienced ratio of merged versus unitary passage?

Answer:
       3:1
We have seen in Chapter 13 that unitary, one-to-one passage has an absolute experienced probability of 0.25, or 25%.  We interpret this as meaning that a person would have a 25% chance of experiencing a unitary passage.
       All other n-to-one passage types add up to a 75% probability.  Since all other n-to-one passages are merged passages, a person should have a 75% chance of experiencing a merged passage.
       The ratio, 75:25, is just 3:1.
       And so the experienced ratio of merged to unitary passage would be 3:1.  Under normal circumstances, a passage participant would be three times as likely to encounter a merged passage as a unitary passage.



Question:
       How likely is ex nihilo passage, relative to participated passage?

Answer:
       equally likely
The rule for predicting the occurrence of n  ex nihilo passages (pex n) is the same as the rule for determining the probability of each n-to-one passage type (pn).  We have seen in Chapter 14 that these probabilities are equivalent:

       pn = pex n = (1/2)n-1    { n = 1 to infinity }

Since the rules are the same, the probabilities are the same.  The probability that a newborn would experience an ex nihilo passage is the same as the probability that the newborn would be the recipient of an n-to-one passage.  And so newborns would be equally likely to experience ex nihilo passage, as not.



Question:
       What is a group's per-generation percentage decrease due to noetic reduction?


Answer:
       50%
The noetic reduction per generation for a particular group is calculated as the sum:

       

We have seen in Chapter 15 that this sum is 50%.  And so with the passing of each generation a given population would undergo noetic reduction into a population just half its original size.
       The cumulative effect would appear to be capable of reducing a whole-species population group down to a single individual, over the course of several hundred years.



These results have been derived by means of informal probability rules.  Because the rules are informal, it is necessary to provide a separate, formal derivation of the results as a double-check of their validity.  As stated previously, the formal derivation has already been done, and is printed in Appendix A.  The formal results are close to the informal results on all points.  The differences are small, and readily accounted for in the computational errors introduced by the particular application of the formal technique.
       These two sets of results are listed side-by-side in Table 16.1 below.  Each row in the table presents the calculated decimal probability for a different passage type.   The final row presents the aggregate merged-to-unitary ratio:

Table 16.1
Comparison of results from two techniques


formal
probability

informal probability
ex nihilo
0.486
0.500
unitary
0.258
0.250
2-to-1
0.265
0.250
3-to-1
0.198
0.188
4-to-1
0.127
0.125
5-to-1
0.074
0.078
merged/unitary
2.85
3.00


The formal and informal results match well.  This satisfies the purely mathematical requirements of the philosophy.
       Beyond this, a visual calculation of the system dynamics can give additional confirmation.  Interactive visuals can engage the reader at each controlled step.  For these reasons I attach a "Monte Carlo program" below.  It performs interactive visual calculations of the three aggregate results reviewed above.

Figure 16.1

Fig. 16.1
Calculator for Metaphysics by Default
(Click image to download the application.)




Appendix B — calculator for Metaphysics by Default.

Appendix C — source code to calculator for Metaphysics by Default.




The calculator of Appendix B sets up a series of random population events.  Once the events have been generated, the program "walks through" the events, progressing along a simulated timeline.  At each time increment the program records the passage events specified by the conditions of that moment. The program then updates the three aggregates derived previously:  the ratio of merged to unitary passage, the ratio of ex nihilo to all other passages, and the noetic reduction percentage.
       A ReadMe file guides installation.  A question mark button on the display panel opens operating instructions. 



The program's numeric output is close to the values of Table 16.1.  It provides a third independent derivation, and confirmation, of the mathematical results.
       As the program runs, its outputs settle near the theoretical values derived informally in Chapters 13-15, and verified formally in Appendix A.  The program contains no explicit rule which forces its calculated results to match the theoretical results.  Instead the cumulative events conform naturally to a probability distribution similar to that found in the formal and informal probability arguments.  For this reason the program produces aggregate values in accord with mathematical prediction.
       Sample Monte Carlo program results, compiled from two different applications,[2] are printed alongside previous results in a combined table, Table 16.2, below.  The close agreement of all three sets of results gives us greater assurance of the soundness of all three techniques.

Table 16.2
Comparison of results from three techniques

Monte Carlo [3]
formal
probability

informal probability
ex nihilo
0.512
0.486
0.500
unitary
0.253
0.258
0.250
2-to-1
0.250
0.265
0.250
3-to-1
0.189
0.198
0.188
4-to-1
0.124
0.127
0.125
5-to-1
0.075
0.074
0.078
merged/unitary
3.14
2.85
3.00



This concludes the mathematical results for Metaphysics by Default.  In our mind's eye we have now stepped onto the fourth of five stepping stones strewn across the river Lethe.  To reach the next and final stone, we will need to consider the metaphysical status of other species.



next    Chapter 17:  Application to Other Species


Chapter 16 Endnotes

[1] A sustained population increase would result in the following qualitative changes to the three aggregate results summarized in this chapter:
  • The merged passage probability would be decreased.
  • The ex nihilo passage probability would be increased.
  • The noetic reduction rate would be decreased.
A sustained population decrease would produce metaphysical changes opposite those attendant a population increase.
[2] The ex nihilo and merged/unitary Monte Carlo ratios were obtained by a Visual Basic 5 program over the course of 1,000,000 simulated events.  Appendix B; Appendix C.  All other Monte Carlo results were obtained by a Macintosh Thin C program over the course of 1,000,007 simulated events.  Appendix D.
[3] Please refer to note 2 for details of the Monte Carlo program results.
 
Copyright © 1999

Wayne Stewart
Last update 4/19/11