Inductive Transformations on Disjoint Orthogonal Systems

Inductive Transformations on Disjoint Orthogonal Systems

Nelson Goodman was a philosopher whose research concerned the area of induction in human thought and its application in science. Induction is the process by which humans, based on observation of previous events, make predictions about future events. The justifying assumption for induction is centered on the idea that when we consider identical events in similar circumstances, we find similar results. Though the hypothesis appears rational, Goodman illustrates that we can induct upon the same system in several different methods to obtain different predictions. We will first outline the paradox obtained and then analyze the paradox to find a potential resolution.

Goodman illustrates the paradox as follows: Assume all emeralds we have seen so far before time t were green. Now let us define a property “grue”  such that an object X is “grue” if and only if the first time it was observed was before time t  and it was green or the first time it was observed was after time t and it was blue. We can write using logical notation as follows:

(⊕ means the “exclusive or” operator)

Observe that it is also the case that all emeralds we have observed before time t so far have also been grue. Ordinarily, with the method of induction we would say that the “next” emerald we observe will be observed to be green. But also observe that it is equally valid to induct on the concept of grue and say that the “next” emerald we observe will be grue. For all emeralds observed before time t, there is no conflict in the predictions, but however, if we predict the color of the first emerald observed after time t, we find that if we induct on green, we predict it to be green, but if we induct on grue, we predict it to be grue and after time t, grue is defined as blue, so our predictions are conflicting.

One might say that because grue is an arbitrary and piecewise definition the contradiction is a result of introducing this term. We could claim to be not as valid as green and blue, but observe that if we define another property “bleen”, that we see that green and blue then become arbitrary and piecewise definitions.

Now observe that green and blue are expressed as the following definitions.

So then it seems that if we take green and blue as our “primitive” structures, then “grue” and “bleen” become our arbitrary definitions, but if we take “grue” and “bleen” as our primitive structures, then blue and green become arbitrary, meaning the resultant inductive paradox allows us to claim equal validity for the next emerald being both green and blue.

To analyze this paradox, let us first fashion some tools with which we can dissect the problem. 

The approach to create a toolset we take requires a few assumptions. The assumptions that we make are as follows, that scientific induction is a “weak” case of proper mathematical induction, and that conceptual changes to the mathematical foundation of induction will have an analogous reflection in scientific induction. These are strong assumptions and are based in a particular interpretation of how constructions of the human mind work. Though we cannot offer “proof” of these assumptions, a little justification can be shown to lend credence to the hypothesis that we make.

Mathematical induction works as follows. We establish that a certain property P holds for a particular natural number q; that is, P(q) is true. This can be thought of as establishing a base case.

We then show mathematically that for an arbitrary natural number k, that P(k) is true implies that P(k+1) must also hold. This means that since we showed the property to hold for q, that it then holds for q+1, which then means that since the property holds for q+1+1, and so on. The scientific analogue to establishing the base case would be that observing that we once found an emerald before and it was green. And for every emerald we found after that one, we found each was green. The analogue to the induction step would be concluding that since we have been finding each consecutive emerald to be green previously, that the next emerald we find will be green and so on and so forth for emeralds after this.

Now we will modify the induction process by adding an additional mathematical framework. Consider that when we induct, we normally do so only on the natural numbers which are a subset of the real numbers.

Observe that the real numbers are a subset of the complex numbers and that another subset of the complex numbers are imaginary numbers. The relationship between imaginary numbers and real numbers is often expressed on a two dimensional Cartesian coordinate plane with the imaginary numbers being the “y” axis and the real numbers being the “x” axis. We will also use this representation of this relationship. Now let us create an imaginary analogue for mathematical induction. 

We establish that a property P holds true for a particular imaginary number g. We then show that for an arbitrary imaginary number j that P(j) is true implies that P(j+i) is true where i=sqrt(-1).

Now to illustrate the consequences and effects of modifying the concept of induction, let us look at the traditional form of induction and the “new” form of induction applied to analogous mathematical properties (this will allow us to extrapolate to the effect on different methods of induction used in scientific analogues).

Consider the formula used to compute the sum of the first n consecutive natural numbers.

To prove this using induction we first verify it for the base case n=1:

1=1(2)/2=1

As we can see the property holds for n=1. Now for the induction step:

So now that we have shown that P holding for an arbitrary value of n implied that P held for n+1,and we showed it to be true for the base case n=1, this means that by induction all natural numbers greater than 1 have the property such that the sum of the consecutive numbers up to that particular number takes the form:

Now with imaginary induction consider the analogous problem: For an imaginary number g what is the sum of consecutive imaginary numbers from i to g? 

We propose that the formula is:

By an almost identical procedure to the one illustrated above, it is easy to show that this indeed is the formula. We italicize the word “almost” because there is a key mutation to the process that is important when considering the scientific analogue of this induction. We observe that the formula that we proposed and verified, and respectively, are fundamentally different, they have analogous structure one could argue, but the formulas themselves are still different.

Since this formula was fundamentally different, this implies that the actual induction step we did is also fundamentally different. If we extrapolate this conceptual difference to the scientific form of induction, specifically with respect to the emerald problem, our two different modes of induction are also fundamentally different.

But however, in the mathematical model, we used a different form of induction on a different set of numbers, but in the scientific problem we used a different form of induction on the same set of emeralds. This means there is anti-symmetry in the scientific problem which we need to untangle.

In order to untangle this anti-symmetry, we properly observe the parallels that exist between the induction step formulas and the induction step definitions of green and blue. As we can write green, blue, “grue”, and “bleen” in terms of each other, we can also do the same for the mathematical formulas.

The significance of these conversion formulas is that they are quite literally a transformation process between the induction steps of each situation upon which we would induct. What is interesting to note is that there is a negative sign in one of these formulas and that is the formula that we would use to convert back from the induction step on imaginary numbers, to real numbers.

Observe that the anti-symmetry that was created between the fact that in the mathematical situation we were using the proper induction step on the set of numbers it was “designed” for, whereas in the emeralds situation, we  are using induction on the same set of emeralds in both places. In order to remedy this anti-symmetry mathematically, when converting back from imaginary to real with the induction step, we simply multiply by i instead of –i. Then we simply end up with the same formula except without the negative sign in front. But observe that when we actually substitute imaginary values of g into it was actually causing the product to be positive before we corrected the formula to reflect the anti-symmetry in the emerald problem.

If we had been inducting on “imaginary” emeralds, then the formula would not need correction, but now that we have corrected the formula, it returns a negative value. Now observe this diagram in which we overlay the complex plane with a graphic representation of green and blue or “grue” and “bleen”:

We see that if we induct by real induction on the “first” emerald after t=0, we arrive arrive at the conclusion that our emerald should be green. This is represented by using this formula and this diagram : 

Imaginary induction on Imaginary emeralds would be represented by this diagram and formula.

Goodman’s “uncorrected” Imaginary induction on real emeralds used the following equation and diagram:

Lastly, our “corrected” Imaginary induction on real emeralds uses the corrected anti-symmetric favoring equation and diagram:

Now we see that both forms of induction result in the same prediction thus the paradox is resolved. The mistake that Goodman made was that he failed to realize that a particular induction step is defined for a particular set of objects on which we are inducting. That is, the base case of emeralds we originally observed doesn’t justify the induction step that Goodman created to use on them. The induction step that Goodman created applies to “imaginary emeralds” and properly applying this on imaginary emeralds results in no problems; however improperly applying the induction step on non-imaginary emeralds results in the contradiction that we observed.

Now that we created and applied the toolset on the new riddle of induction, our analysis has concluded that the paradox is resolved.

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