54 3 A Patternist Philosophy of Mind
This degree is called the pattern intensity of P in X. It quantifies the extent to which P is a pattern in X. Supposing that F(P) = X, then the first factor in the definition equals 1, and we are left with only the second term, which measures the degree of compression obtained via representing X as the result of P rather than simply representing X directly. The greater the compression ratio obtained via using P to represent X, the greater the intensity of P as a pattern in X. The first time, in the case F(P) #4 X, adjusts the pattern intensity downwards to account for the amount of error with which F(P) approximates 4 X. If one holds the second factor fixed and thinks about varying the first factor, then: The greater the error, the lossier the compression, and the lower the pattern intensity.
For instance, if one wishes one may take c to denote algorithmic information measured on some reference Turing machine, and F(X) to denote what appears on the second tape of a two-tape Turing machine ¢ time-steps after placing X on its first tape. Other more naturalistic computational models are also possible here and are discussed extensively in Appendix 1 of [Goe06a].
Definition 2 The structure of X € M is the fuzzy set Stx defined via the membership function Xstx(P) =e
This lets us formalize our definition of “mind” alluded to above: the mind of X as the set of patterns associated with X. We can formalize this, for instance, by considering P to belong to the mind of X if it is a pattern in some Y that includes X. There are then two numbers to look at: .% and P(Y|X) (the percentage of Y that is also contained in X). To define the degree to which P belongs to the mind of X we can then combine these two numbers using some function f that is monotone increasing in both arguments. This highlights the somewhat arbitrary semantics of “of” in the phrase “the mind of X.” Which of the patterns binding X to its environment are part of X’s mind, and which are part of the world? This isn’t necessarily a good question, and the answer seems to depend on what perspective you choose, represented formally in the present framework by what combination function f you choose (for instance if f(a,b) = a™b?” then it depends on the choice of 0 <r < 1).
Next, we can formalize the notion of a “pattern space” by positing a metric on patterns, thus making pattern space a metric space, which will come in handy in some places in later chapters:
Definition 3 Assuming M is a countable space, the structural distance is a metric dst defined on M via
dst(X,Y) =T(xstx,Xsty) where T is the Tanimoto distance. The Tanimoto distance between two real vectors A and B is defined as 7 A-B Al? + (|B? -— A: B
and since M is countable this can be applied to fuzzy sets such as Stx via considering the latter as vectors. (As an aside, this can be generalized to uncountable M as well, but we will not require this here.)
T(A, B)
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