I have a strong belief that life is enriched with projects. I also have an equally strong trust in self-awareness. However, my self-awareness has been fairly well entrenched in the rational and historically scant of the emotional.

Therefore, over the course of the last month, I set out to become more aware of my emotions. I have been doing this by keeping a journal specifically devoted to the exploration of my feelings, reading books on the topic [1,2] and undertaking a course of 12 sessions of psychotherapy, of which, at the time of writing, I have completed two.

I expect to make many profound (to me) realizations about myself, but I want to share with you, one counter-intuitive discovery so far.

Being "nice" isn't fair. In the name of playing "nice", many good relationships have been ruined. Relationships with real people in the real world are not ideal. This point is probably so tautologically obvious it needs no further exposition. When people get mad, they can feel hatred. But this hatred is not incompatible with love. Quite the contrary, it is just another side of it. We sometimes think that if we swallow these feelings of bitterness, our relationship will be more "ideal". I have swallowed negative feelings often hoping that it will be a boon for my relationships. What one learns though, upon reflection, is that the non-expression of the negative eventually leads, quite naturally, to a non-expression of the positive.

The great lesson of psychotherapy is that the converse is also true. By expressing negative emotions from past events, positive feelings return. Rollo May has this to say about it:

"A curious thing which never fails to surprise persons in therapy is that after admitting their anger, animosity, and even hatred for a spouse and berating him or her during the hour, they end up with feelings of love toward this partner. A patient may have come in smoldering with negative feelings but resolved, partly unconsciously, to keep these, as a good gentleman does, to himself; but he finds that he represses the love for the partner at the same time as he suppresses his aggression." [2]

May goes on to note that this denial is typical in "our bourgeois society." The prognosis for people suffering from such emotional repression is not good - leading eventually to a "feeling [of] no connection and no vital drive toward the other person".

As I promised at the outset of this blog, I expect to update you on my life and my evolution as a person, and I look forward to those updates in the future (as I hope you do). And I plan to follow the first rule of Project Emotions: "to project emotions" (in blog or in relationship). Please remember, this relationship - if that term is appropriate - must be reciprocal to survive, so please let me know what you think!

[1] Irvin Yalom "Love's Executioner"

[2] Rollo May "Love & Will"

## Wednesday, August 19, 2009

## Wednesday, August 12, 2009

### The meaning of "meaning"

Does meaning exist in our minds or is it something that can be substantiated in the physical world?

This is a big question and philosophers have struggled with it for a long time. The philosophical problem that I am asking is, "Are there symbols that have inherent meaning, or are symbols imbued with meaning ex post facto?" Is meaning intrinsic or extrinsic? Is it layered onto things or is it natively there?

In the extreme physical realist interpretation of reality, the universe has no intrinsic meaning and the only thing that persists after observers stop observing is matter. In such a universe, all meaning, text and subtext is non-existent.

Duality is the atom of meaning

Notice what happens though when one introduces thought into a universe. Suddenly, relationships between masses in the universe become apparent to minds with pattern recognition capabilities. And now, one can begin to ascribe meaning to observations. In my opinion, the critical feature of meaning is perceiving duality in an object. Note, that I have not said anything about the realism of this duality, just that it must be perceived. Duality is a pre-requisite for meaning. Comprehension of the physical nature of the event and the higher level meaning must coexist in an observer's mind simultaneously.

I think a few examples will help illustrate my point. The observation that mass, as such, exists in the universe is not enough for meaning. Seeing the thing is not sufficient. A phagocyte "sees" a bacteria invader and engulfs it, but I would hesitate to claim that it attributes some meaning to the engulfing or experiences it on any other level. Similarly, a thermostat "perceives" that it is "too cold," but could we honestly claim that it attributes any meaning to the phenomenon?

I would argue that in the above examples, we must say that those may be interesting relationships to an observer, but the objects in the system relate to one another in a physical sense and do not invoke any higher level relationships as far as they are concerned.

What do we mean when we claim "meaningfulness"?

The critical point in this discussion is that while we of course must follow the rules of physical objects since we ourselves are physical in nature, the really interesting stuff, the stuff of consciousness and humanity, occurs at a higher level. That is to say, while we can perceive the physical reality of a mark on a page, or the digital readout on a thermostat, meaning is derived from the fact that we perceive simultaneously the dual nature of the mark or the readout and the semantic content of the message.

Does duality exist in the physical reality of nature?

Here is where it starts to get pretty interesting. We certainly know from double-slit experiments that electrons, photons and other quantum objects have duality. This illustration, by Douglas Hofstadter, is the best that I know that artistically captures the multifaceted relationship between the perception of duality by an observer of 'light' as both a 'particle!' and a 'wave!'!*

The fascinating thing here is related to the sheer number of ways this statement can be read. As read by an English speaking observer, the message has at least three levels. 1. As marks on a page, 2. As a sentence about light being a wave and 3. As a sentence about light being a particle. But there are more! 4. The observer can experience this as a statement about the duality of light. 5. She can experience it as an analogy about the true nature of light, which has a certain existence initially (as just marks on a page) until an observer, much like herself, makes an observation and the marks become a particle or wave or both. 6. Now inside the message (via 5), she can see the statement as a commentary on the nature of observation and art and meaning and it expands from here!

We can go on from here, but it is interesting to note that once multiple levels are allowed/perceived, as they must be for meaning to be derived in the first place, that very over-determinism, which is a pre-requisite for meaning itself, necessarily obfuscates the message's meaning.

* I obtained Douglas Hofstadter's drawing from the wikipedia entry for ambigrams. These are amazing pieces of art, for which, Douglas Hofstadter coined the term. I hope this short essay will be an impetus for you to check out ambigrams further and read one of his books that feature them. I recommend Metamagical Themas and Gödel Escher Bach.

This is a big question and philosophers have struggled with it for a long time. The philosophical problem that I am asking is, "Are there symbols that have inherent meaning, or are symbols imbued with meaning ex post facto?" Is meaning intrinsic or extrinsic? Is it layered onto things or is it natively there?

In the extreme physical realist interpretation of reality, the universe has no intrinsic meaning and the only thing that persists after observers stop observing is matter. In such a universe, all meaning, text and subtext is non-existent.

Duality is the atom of meaning

Notice what happens though when one introduces thought into a universe. Suddenly, relationships between masses in the universe become apparent to minds with pattern recognition capabilities. And now, one can begin to ascribe meaning to observations. In my opinion, the critical feature of meaning is perceiving duality in an object. Note, that I have not said anything about the realism of this duality, just that it must be perceived. Duality is a pre-requisite for meaning. Comprehension of the physical nature of the event and the higher level meaning must coexist in an observer's mind simultaneously.

I think a few examples will help illustrate my point. The observation that mass, as such, exists in the universe is not enough for meaning. Seeing the thing is not sufficient. A phagocyte "sees" a bacteria invader and engulfs it, but I would hesitate to claim that it attributes some meaning to the engulfing or experiences it on any other level. Similarly, a thermostat "perceives" that it is "too cold," but could we honestly claim that it attributes any meaning to the phenomenon?

I would argue that in the above examples, we must say that those may be interesting relationships to an observer, but the objects in the system relate to one another in a physical sense and do not invoke any higher level relationships as far as they are concerned.

What do we mean when we claim "meaningfulness"?

The critical point in this discussion is that while we of course must follow the rules of physical objects since we ourselves are physical in nature, the really interesting stuff, the stuff of consciousness and humanity, occurs at a higher level. That is to say, while we can perceive the physical reality of a mark on a page, or the digital readout on a thermostat, meaning is derived from the fact that we perceive simultaneously the dual nature of the mark or the readout and the semantic content of the message.

Does duality exist in the physical reality of nature?

Here is where it starts to get pretty interesting. We certainly know from double-slit experiments that electrons, photons and other quantum objects have duality. This illustration, by Douglas Hofstadter, is the best that I know that artistically captures the multifaceted relationship between the perception of duality by an observer of 'light' as both a 'particle!' and a 'wave!'!*

The fascinating thing here is related to the sheer number of ways this statement can be read. As read by an English speaking observer, the message has at least three levels. 1. As marks on a page, 2. As a sentence about light being a wave and 3. As a sentence about light being a particle. But there are more! 4. The observer can experience this as a statement about the duality of light. 5. She can experience it as an analogy about the true nature of light, which has a certain existence initially (as just marks on a page) until an observer, much like herself, makes an observation and the marks become a particle or wave or both. 6. Now inside the message (via 5), she can see the statement as a commentary on the nature of observation and art and meaning and it expands from here!

We can go on from here, but it is interesting to note that once multiple levels are allowed/perceived, as they must be for meaning to be derived in the first place, that very over-determinism, which is a pre-requisite for meaning itself, necessarily obfuscates the message's meaning.

* I obtained Douglas Hofstadter's drawing from the wikipedia entry for ambigrams. These are amazing pieces of art, for which, Douglas Hofstadter coined the term. I hope this short essay will be an impetus for you to check out ambigrams further and read one of his books that feature them. I recommend Metamagical Themas and Gödel Escher Bach.

## Thursday, August 6, 2009

### Problems, solutions and verification

Puzzles are fun. But what is a puzzle really. Puzzles are a class of problems that have been solved before. In this essay, I want to draw your attention to three classes of problems: puzzles, near-puzzles and theories.

The first class comprises the sort of problems for which a known algorithm exists to get us to the solution. These types of problems have the property that as soon as we identify their type (i.e. that it is a puzzle like one I have already solved), we just need to apply the correct formula and eventually we will get the answer. Let's call the class of these puzzle type problems, P.

The second class of problems are problems, for which, while no algorithm may exist to reach a solution, if a solution is offered, it could be tested and shown to be a solution. Since these problems are similar to puzzles, we'll call the class of near-puzzle type problems, NP.

There is a third class of problems. These are the problems that have not been solved, and/or if a solution existed, the solution could not be verified. This class of problems are similar to theories, and therefore we'll call the class of theory-type problems, T.

Some examples

Summation. Summation is a kind of "puzzle." The summation puzzle asks, "For any two numbers, what number is their resulting sum?" We know that this puzzle is a P-type problem since there is a strict rule for summation of any two numbers and the rule always generates the answer.

Temporal questions. One cannot predict the future. I can be pretty sure, but I can't be certain about it until it has safely become the past. So problems like what will I have for lunch tomorrow is a NP-type problem. I may think it will be a peanut butter and jelly sandwich, but until I have it, I can't be sure. Of course, these solutions can be verified simply by waiting long enough.

Theory verification. Some problems are problems that relate to the truth of general theories. Theory verification has been known to be a general problem for a long time in the philosophy of science. Given any scientific theory there is no way to prove that it is true. For example, if I have a theory that has some explanatory power, such as, "Evolution via natural selection is an explanatory principle for the myriad complex living organisms that exist today," it pertains as an explanation until it doesn't. That is, until I find enough scientific data that debunk it (such as serious inconsistencies in the fossil record, for example) I can believe it to be true. This is true of all scientific theories, even the historically unexceptionable ones like quantum mechanics or the modern evolutionary synthesis.

Is the Goldbach Conjecture P, NP or T?

GC: "Every even integer greater than 2 can be written as the sum of two primes"

Let's presume, as Euler may have, that the GC is true but not provable. This conjecture would belong to the T-type problems.

However, if it were false, then it would presumably belong to the NP class (would be verifiable), and by extension, belong to the P class since the counter-example would be a finite number and an algorithm that checked every even integer greater than 2 would be a valid algorithm.

Some interesting properties of P's and NP's.

You may have noticed that the class of P problems are very similar to NP problems, in some regards. Unlike T problems, an NP problem is similar to a P problem, in that, given its solution, it can be verified. We could certainly imagine mistaking a P-type problem for an NP-type problem in cases where we didn't notice that it was the same riddle in a different frame, figuratively speaking. Another interesting feature of P-type problems is that it is not inconsistent to classify them as NP problems. All P-type problems are NP-type problems. Notice, if the reverse is true, then near-puzzles would be puzzles.

Given all the similarities, we may begin to suspect that, in actual fact, P=NP. It is at least possible that for all those NP-type questions, they are really P-type questions, and it is our own limited knowledge that has allowed us to initially misclassify them. How could we find out? Does P=NP? That is an interesting problem. Is that problem, itself, a P-, NP- or T-type problem?

Classifying "P=NP?"

Assuming that "P=NP?" is one of the three, it is either the case that "P=NP?" is a P- (and therefore also NP), NP- or T-type problem.

Let's write out the question completely. The question we are asking is, "Is the problem 'Are all puzzle type problems the same as all near-puzzle type problems?' a member of P or NP or T?"

It is possible that the question does not belong to P or NP (case: not P, not NP).

In this case, the question could belong to the T-type questions. If that were the case, then there would be no way to satisfactorily answer "P=NP?". It is possible that it could be true that P=NP, but there would be no way to prove it. This is similar to the incompleteness result guaranteed in sufficiently complex (and consistent) formal systems as explained by Gödel.

Let's presuppose that there is a solution to this problem and set aside the possibility that the general question is a T-type problem. The following two possibilities would remain ([not P but NP] or [P and NP])

It is possible that the question does not belong to P but does belong to NP (case: not P but NP).

In this case, it is clear that the underlying question "P=NP?" would need to be false since if it were the case that P=NP, the above presupposition (not P but NP) would be meaningless. Assuming that the above presupposition is not meaningless, it must be the case that P≠NP. However, it is not consistent for the question "P=NP?" to be a member of NP (and not P) and to show that P≠NP. For, if we were able to definitively show that P≠NP, then the question "P=NP?" must be a member of P and NP (the solution algorithm thereby classifying it as P would of course be this proof!).

Continuing to assume that the question is not a T-type question, it follows that the only choice that remains is that the question belongs to P and (trivially) NP (case: P and NP).

If the problem "P=NP?" is a member of P, this means that the question "Are all puzzles and near-puzzles equivalent?" has a solution and there is an algorithm to demonstrate it! Either P=NP or P≠NP.

If P=NP, we must be capable of imagining an algorithm existing, which can rule out any problems with algorithmically verifiable solutions, for which, no algorithm can provide the solution (an algorithm exists and it rules out NP-type problems that are not also P-type). This should be an easy task since we have already assumed that there exists an algorithm that has solved the harder problem "P=NP?" It is a contradiction in terms to suppose that an algorithm that can verify solutions cannot also generate solutions themselves. For example, the algorithm, try all potential solutions one at a time would be a viable algorithm for any problem one can suppose.

Notice that since we have assumed that there is a solution to the problem (that it is not a T-type problem) we have implicitly eliminated all true but unprovable claims from the pot of questions that are valid. Under these premises we are assuming that an algorithm doesn't get stuck checking the types of problems that would ultimately be T-type problems. If we know a priori which types of problems these are, then we guarantee that all our searches for solutions will terminate, and in that case, the search space is finite. This guarantees that P=NP.

For similar reasons as above, we have ruled out the possibility that P≠NP. For that to be true, it must be possible for there to exist an algorithm that can identify at least one problem for which solutions, if found, could be verified and further, the algorithm must show that for this problem there exists no algorithm one might use to find a solution. Further, this algorithm must be able to reconcile this fact with the fact that the more general problem is P=NP? is itself a P-type problem. Just as before, there is no way that P≠NP is a solution given the particular search space to which we have confined ourselves.

Conclusions

I have shown that either "P=NP?" has no solution (i.e. it is a theory type question) or P=NP. I have purposefully used the computer science terminology in their own P vs. NP debate because I believe the argument above is at least useful as an analogy. Unlike the computer science debate, the above has not made specific mention of an algorithm's time scaling with respect to a given input, which is of course a very important detail. For example, algorithms with exponential time solutions would still be members of P according to the above argument, while they would be excluded from P in the computer science version of the argument.

The first class comprises the sort of problems for which a known algorithm exists to get us to the solution. These types of problems have the property that as soon as we identify their type (i.e. that it is a puzzle like one I have already solved), we just need to apply the correct formula and eventually we will get the answer. Let's call the class of these puzzle type problems, P.

The second class of problems are problems, for which, while no algorithm may exist to reach a solution, if a solution is offered, it could be tested and shown to be a solution. Since these problems are similar to puzzles, we'll call the class of near-puzzle type problems, NP.

There is a third class of problems. These are the problems that have not been solved, and/or if a solution existed, the solution could not be verified. This class of problems are similar to theories, and therefore we'll call the class of theory-type problems, T.

Some examples

Summation. Summation is a kind of "puzzle." The summation puzzle asks, "For any two numbers, what number is their resulting sum?" We know that this puzzle is a P-type problem since there is a strict rule for summation of any two numbers and the rule always generates the answer.

Temporal questions. One cannot predict the future. I can be pretty sure, but I can't be certain about it until it has safely become the past. So problems like what will I have for lunch tomorrow is a NP-type problem. I may think it will be a peanut butter and jelly sandwich, but until I have it, I can't be sure. Of course, these solutions can be verified simply by waiting long enough.

Theory verification. Some problems are problems that relate to the truth of general theories. Theory verification has been known to be a general problem for a long time in the philosophy of science. Given any scientific theory there is no way to prove that it is true. For example, if I have a theory that has some explanatory power, such as, "Evolution via natural selection is an explanatory principle for the myriad complex living organisms that exist today," it pertains as an explanation until it doesn't. That is, until I find enough scientific data that debunk it (such as serious inconsistencies in the fossil record, for example) I can believe it to be true. This is true of all scientific theories, even the historically unexceptionable ones like quantum mechanics or the modern evolutionary synthesis.

Is the Goldbach Conjecture P, NP or T?

GC: "Every even integer greater than 2 can be written as the sum of two primes"

Let's presume, as Euler may have, that the GC is true but not provable. This conjecture would belong to the T-type problems.

However, if it were false, then it would presumably belong to the NP class (would be verifiable), and by extension, belong to the P class since the counter-example would be a finite number and an algorithm that checked every even integer greater than 2 would be a valid algorithm.

Some interesting properties of P's and NP's.

You may have noticed that the class of P problems are very similar to NP problems, in some regards. Unlike T problems, an NP problem is similar to a P problem, in that, given its solution, it can be verified. We could certainly imagine mistaking a P-type problem for an NP-type problem in cases where we didn't notice that it was the same riddle in a different frame, figuratively speaking. Another interesting feature of P-type problems is that it is not inconsistent to classify them as NP problems. All P-type problems are NP-type problems. Notice, if the reverse is true, then near-puzzles would be puzzles.

Given all the similarities, we may begin to suspect that, in actual fact, P=NP. It is at least possible that for all those NP-type questions, they are really P-type questions, and it is our own limited knowledge that has allowed us to initially misclassify them. How could we find out? Does P=NP? That is an interesting problem. Is that problem, itself, a P-, NP- or T-type problem?

Classifying "P=NP?"

Assuming that "P=NP?" is one of the three, it is either the case that "P=NP?" is a P- (and therefore also NP), NP- or T-type problem.

Let's write out the question completely. The question we are asking is, "Is the problem 'Are all puzzle type problems the same as all near-puzzle type problems?' a member of P or NP or T?"

It is possible that the question does not belong to P or NP (case: not P, not NP).

In this case, the question could belong to the T-type questions. If that were the case, then there would be no way to satisfactorily answer "P=NP?". It is possible that it could be true that P=NP, but there would be no way to prove it. This is similar to the incompleteness result guaranteed in sufficiently complex (and consistent) formal systems as explained by Gödel.

Let's presuppose that there is a solution to this problem and set aside the possibility that the general question is a T-type problem. The following two possibilities would remain ([not P but NP] or [P and NP])

It is possible that the question does not belong to P but does belong to NP (case: not P but NP).

In this case, it is clear that the underlying question "P=NP?" would need to be false since if it were the case that P=NP, the above presupposition (not P but NP) would be meaningless. Assuming that the above presupposition is not meaningless, it must be the case that P≠NP. However, it is not consistent for the question "P=NP?" to be a member of NP (and not P) and to show that P≠NP. For, if we were able to definitively show that P≠NP, then the question "P=NP?" must be a member of P and NP (the solution algorithm thereby classifying it as P would of course be this proof!).

Continuing to assume that the question is not a T-type question, it follows that the only choice that remains is that the question belongs to P and (trivially) NP (case: P and NP).

If the problem "P=NP?" is a member of P, this means that the question "Are all puzzles and near-puzzles equivalent?" has a solution and there is an algorithm to demonstrate it! Either P=NP or P≠NP.

If P=NP, we must be capable of imagining an algorithm existing, which can rule out any problems with algorithmically verifiable solutions, for which, no algorithm can provide the solution (an algorithm exists and it rules out NP-type problems that are not also P-type). This should be an easy task since we have already assumed that there exists an algorithm that has solved the harder problem "P=NP?" It is a contradiction in terms to suppose that an algorithm that can verify solutions cannot also generate solutions themselves. For example, the algorithm, try all potential solutions one at a time would be a viable algorithm for any problem one can suppose.

Notice that since we have assumed that there is a solution to the problem (that it is not a T-type problem) we have implicitly eliminated all true but unprovable claims from the pot of questions that are valid. Under these premises we are assuming that an algorithm doesn't get stuck checking the types of problems that would ultimately be T-type problems. If we know a priori which types of problems these are, then we guarantee that all our searches for solutions will terminate, and in that case, the search space is finite. This guarantees that P=NP.

For similar reasons as above, we have ruled out the possibility that P≠NP. For that to be true, it must be possible for there to exist an algorithm that can identify at least one problem for which solutions, if found, could be verified and further, the algorithm must show that for this problem there exists no algorithm one might use to find a solution. Further, this algorithm must be able to reconcile this fact with the fact that the more general problem is P=NP? is itself a P-type problem. Just as before, there is no way that P≠NP is a solution given the particular search space to which we have confined ourselves.

Conclusions

I have shown that either "P=NP?" has no solution (i.e. it is a theory type question) or P=NP. I have purposefully used the computer science terminology in their own P vs. NP debate because I believe the argument above is at least useful as an analogy. Unlike the computer science debate, the above has not made specific mention of an algorithm's time scaling with respect to a given input, which is of course a very important detail. For example, algorithms with exponential time solutions would still be members of P according to the above argument, while they would be excluded from P in the computer science version of the argument.

Labels:
Gödel,
Goldbach,
Incompleteness,
P vs. NP,
Philosophy,
puzzles,
verification

Subscribe to:
Posts (Atom)