Friday, July 6, 2012

3. Induction

Part of my Subjective Epistemicism series

So far, we have managed to demonstrate that the self ("I") and reality ("This") exist. We've also mentioned that they combine as aspects of a single phenomenon ("existence" or "being.") But neither that proposition, nor the definition of "I" (which is only philosophically negative in terms of its properties) will take us much further in developing a worldview. All else going forward depends on entirely the proven basis of reality: "This," the basis of all empirical fact. Having supported the admittance of empirical fact, we will now explore how we might evaluate its contents.

3a. Experience ("This") displays certain consistencies

Imagine what a perfectly random experiential milieu might look like. Perhaps you might picture an visual field of static and an auditory field of white noise combined with an electrical prickly feeling all over your body. Or you might imagine a field of constantly blending colors and vague sounds combined with the feeling of bugs crawling all over your skin, such as might be provided by an extremely chaotic hallucinogenic "bad trip." The latter image is more likely to represent randomness in a neurological system, but either (or any other scheme of pure empirical randomness one could propose) will suffice.

I can personally report that this is not my experience of reality. Anyone who can understand what I'm saying at this moment must admit that it is not theirs, either.

To the degree that reality is not random, we must admit that it displays certain consistencies. What those consistencies consist of we'll save for later. What is important to establish now is that reality is something other than random. That it behaves in certain consistent fashions.

From whatever those consistencies may involve, we can begin to extrapolate some theories about how reality behaves. Such theories merely amount to descriptions of observed consistencies. They have no truth-value in and of themselves; they are merely ways of talking about observed consistencies in the universe's behavior.

Only some greater-than-zero amount of consistency is required in order for descriptions of consistency to be valuable. We don't need a perfectly consistent universe in order to discuss even partial consistencies that we have observed. Even a consistency that amounts to a mere tendency is useful. It is as far from a perfectly random universe as anything else. We need not describe a perfectly consistent universe in order to find a partial description of consistency useful.

In fact, we can never know with certainty that our universe is perfectly consistent. It may have heretofore been so, but that doesn't preclude the possibility that it might surprise us at any moment.

3b. The problem and principle of induction.

If we are to be able to justify anything further than merely describing our observations in the moment and expecting nothing to follow from those descriptions - in fact if we are to justify ever inferring or doing anything at all - then we must find some kind of support for the principle of induction: that unobserved instances (such as future events) are likely to follow the pattern of observed instances (such as past experiments).

However, the principle of induction is problematic to philosophy. David Hume provided the best and most complete formulation of this problem, pointing out that the trouble with assuming such a principle is that A) it cannot be inferred from experience, because it itself is assumed as an implicit premise in any inference from experience; and B) it does not qualify for a principle that could be adopted as a priori, since its negation does not produce a contradiction. Thus, from the position of logical positivism (which allows only those two sources of knowledge) there can be no reason to believe that the universe should continue to appear consistent in any given way, even though it always has.

In addition to the difficulties raised by Hume, there's another obstacle that bears mentioning. That is, were someone able to provide a deductive formal argument supporting the principle of induction, the argument constructed would ultimately constitute circular reasoning.

This is because the only reason that we adopt formal deductive logic at all is because it works. If it didn't work, then we wouldn't claim it to be the most effective method of determining validity. When one asks why we should use formal logic at all, ultimately the best (and only reasonable) argument to reply with is to point at the many ways in which it has proven effective throughout the history of human thought in eliminating falsehoods and exposing truths. After all, it would hardly be reasonable to provide a formal logical argument for why formal logic should be employed to begin with. And as logical positivism allows only inference from experience as a remaining source of knowledge, we must admit that we infer the validity of the deductive method from experience. As Hume points out, any such inference assumes the principle of induction.

So we are already assuming the inductive principle whenever we advocate logical argumentation. Which is to say, deductive logic rests upon induction. To provide a sound deductive proof for the inductive principle (when deductive proof itself follows from induction) would thus constitute a special case of self-justifying argument, the conclusion of which supports not only its own premises, but also the very method used to assemble the argument.

When observing that induction causally precedes deduction, it becomes far less of a "problem" that we cannot deductively prove induction. It's just the state of affairs in which things are ordered according to epistemic causality, and it would be nonsensical to complain that it should be otherwise.

Even if we can't provide an argument for the principle of induction, what we can do is compare it to its negation, and discuss the contrast. Although the contrary position doesn't meet the a priori requirement of logical self-contradiction, there are some things that can be said about it. Adherence to logical positivism adopts only the kind of truth values that deductive argumentation produces. However, there are other criteria by which a principle might be judged which can be said to be reasonable, yet which do not involve such absolute truth values. These include: utility, likelihood, and being non-arbitrary.

3c. The negation of the inductive principle provides no path to utility.

Everyone already acts as if the principle of induction were useful. Anyone who disagrees with it would be a hypocrite to say so in words (much less while wearing clothes and in the presence of their debate partner), since they would have no reason to expect speaking words to be of any use. The only intellectually honest way to argue against induction would require dropping to the ground, twitching and drooling, because anyone who believed that it held no utility would have to act as if the universe were random, which is to say, act completely randomly.

There are a practically unlimited number of things that I could do at any moment. Of those things, only a few of them cohere with the consistencies of the universe that I've observed, i.e. follow the principle of induction. To deny induction is to do something other than to behave as if the universe is (or should continue to be) at all consistent. It is to behave randomly.

Thought experiment:

You are led into an empty room. Upon one wall is a big red button next to a dispensing chute.

I ask you to press the big red button. You do. A white ping-pong ball comes out of the chute.

I offer you an even bet of $5 that you cannot predict what will come out of the chute next. You decline the bet, as well you should; the number of things that could have come out of the chute are theoretically unlimited: a white ping-pong ball, a black one, a red one, a set of car keys, a feather, or nothing at all.

I ask you to press the big red button. You do. Again, a white ping-pong ball comes out of the chute. Again, I offer you an even bet of $5 that you cannot predict what will come out of the chute next. You decline the bet. As well you might; the second case only matches the first. At best (given only a minimum of two possibilities) the odds are 50/50 that the second item matched the first. The first only set the baseline for comparison for the second.

We repeat this process nine more times. Each time you press the big red button, a white ping-pong ball emerges from the chute. At this point, the maximum odds of ten items matching a given initial baseline are at most one in 210, or 1:1,024. That's the formula for only two possibilities; if the number of possibilities to be considered are 3 or more, the odds of the prior sequence having occurred randomly are even more slim (310, 410, etc).

Again, I offer you an even bet of $5 that you cannot predict what will come out of the chute next. The odds that you would lose this bet (i.e. that the results were, and therefore will be random) are at minimum 1:1,024.

If you still decline the bet, let's have you press the big red button ten more times. Each time, a white ping-pong ball comes out of the chute. Now the odds of that being a random result are at least one in 220, or 1:1,048,576.

Again, after all of this, I offer you an even bet of $5 on the next outcome. Even if you still decline, there must be some point in the test (30 white ping-pong balls? 40? 100?) at which you should take the bet, or else any reasonable third party observer should judge you as being either overly obstinate, merely irrational, or entirely insane (per the old saw "doing the same thing over and over again and expecting different results").

Generally speaking, anyone who behaves with respect to their experiential universe as if they expect a predictable outcome from it, is "taking the bet." Any time that you behave as though you expect that what is going to happen is consistent with your past experience, you are "taking the bet" of risk and reward that is always and every moment offered to you by the universe. In fact, any willful action that a sane person ever takes falls in this category.

However, it's not enough to merely point out that we all do follow the principle of induction. What we're looking for is why we should. Or at least looking for some feature that distinguishes it from its competitors.

Although I can't assert with certainty that the universe will behave consistently as it has to date, no other pattern of behavior even suggests how it might give me useful results.

I could come up with some alternative principle such as "always bark when you see red," or "constantly spin to the left." Not only is there no more logical reason to follow such a principle as that of induction, such principles don't even suggest how they might lead to any desired outcome whatsoever. They all equally rely on the remote chance that in following one, I might occasionally stumble into a desired outcome by purely random chance. Furthermore the negation of induction suggests that for any such principle, the odds of a desired outcome are all exactly equal. That is to say, neither barking nor spinning strategies have any basis to recommend themselves with respect to the other. They are both predicated on the notion that all strategies provide an exactly equally likelihood of positive outcome, that being entirely random.

In contrast, induction does suggest a mechanism regarding how it might lead to a desired outcome; if the universe remains consistent, then I can derive a course of behavior that would lead to a better outcome than otherwise. It is the case that the universe might not remain consistent. But at least induction has the merit that there may be some relationship between my behavior and achieving a desired outcome. No competing behavior-guiding principle can make the same claim. In fact, they must all deny it.

When given a choice between a behavioral principle that suggests "there may be some relationship between my behavior and a desired outcome," and a theoretically infinite number of behavioral principles that assume "there is no relationship between my behavior and a desired outcome," it can never be to my detriment to behave according to the first principle. Furthermore, it may be to my benefit. None of the other principles can say the same. Therefore, I should follow the first principle.

3d. The negation of the inductive principle is immensely unlikely.

As a more general (but parallel) version of the observation that random behavior is unlikely to produce desired outcomes, let's see if we can determine what the likelihood is that the universe is actually random.

To do this, we need to look at the ratio of consistent universes to all possible universes given the same number of discrete events, and consider where that ratio lands for the number of events in the universe in which I exist. Mind you, I'm not comparing the odds of exactly this universe occurring from random chance; such an approach would betray a certain prejudice for the world as I know it. Rather, I'm looking for the odds of any universe which displays consistency.

Let's go back to the room, the big red button and the chute. For purposes of this version of the thought-experiment, let's assume that the chute only provides either a white ping-pong ball or a black one.

  • After pressing the button twice (n=2, where n is the number of events or results) I have either two white ping-pong balls or two black ones, or one of each. To claim that a consistent (i.e. repeating) pattern had been discovered, it must have occurred more than once. Therefore, two white ping-pong balls or two black ping-pong balls can be said to establish consistent patterns, out of the 4 combinations available. Thus at n=2, the odds that an apparently consistent (but actually coincidental) universe are 2 consistencies in 4 possibilities, or 1:2.
  • At n=4, we can add white/black (alternating) to the discovered patterns. However, black/white (alternating) should be counted as the same pattern, only shifted over to the left or right by one instance or digit; that is to say, white-black-white-black is the same pattern as black-white-black-white. What we're looking for is new patterns, so these both count as 1 discovered pattern. Thus at n=4, we've discovered one more consistent pattern, and the odds of coincidence are now 3:16
  • At n=6, we can add white/black/black and black/white/white. The odds are now 5:64.
  • At n=8, we can add white/white/black/black, white/black/black/black, and black/white/white/white. The odds are 8:256, or 1:32.

As we can see, with every increase in n, more possible repeating (or consistent) patterns become discovered. That increase in discovered constistencies expands at a roughly geometric rate with every increase in n. At every even n, we gain 2 new consistent patterns, +1 for every n that is a power of 2.

To be complete, we might also mention that there are a certain fixed number of patterns which are constant for mathematics and which don't appear with increasing n's, such as the prime number series, the Fibonacci series, etc. However, since the number of thse don't increase with n, they become immediately insignificant.

On the other hand, the total number of possibile universes against which such consistent patterns can be compared amounts to 2n. Which is to say, the total number of possibilities increases by an exponential (rather than geometric) rate. Thus, as the number of events increases, the number of consistent possibilities becomes a vastly smaller ratio of the total number of random possible outcomes. Whereas the number of consistent patterns increases gradually (although a greater number of patterns are added with each increase in n) the number of possible random results races away at a vastly accellerating rate.

Also, I should now mention that we've only been considering two possibilities per event above. The number of possibilities per event for the real world is considerably more than the exponential base of 2 that I've been using for the purpose of illustration. While a greater (3 or more) possibility-base does expand the number of potential patterns discovered, it vastly increases the curvature of the number of possibilities with each increase in n. Which is to say, considering that there are a theoretically unlimited number of possibilities for each event only increases the ratio to unimaginable levels.

We might nitpick about precisely what n is quantified as for my subjective state of being. Perhaps it should be left to neurology to find something like an approximate measurement. But there certainly must be more than a few dozen "events" which make up my understanding of the world as it exists in my mind. And a few dozen is easily enough to make the case that the odds against my experiential universe being as consistent as it has been presented to me, are immense. For every moment of my life, the average number of possible outcomes to the power of the number of such moments are the odds against me having been able to find any consistent patterns within my life at all.

And yet, that I have randomly overcome such odds is exactly what the negation of the principle of induction asserts. It asserts that the consistent universe that I have lived in and dealt with all this time has only presented itself as consistent due to a colossal chain of unlikely coincidences. That the true subjective state of being constitutes flailing about in a random universe of constantly shifting meaningless sensation, and I was the one lucky jackpot winner who has (so far) experienced what has looked like a universe governed by some form of cause and effect. But (per this claim) every observation of consistency that I've made in my entire life has actually been a sheer coincidence. Every moment has coincidentally behaved - through sheer random chance - as if it had been consistent with the entirety of my previous life. As if it were governed by some type of causal principle, but was not.

Consider the magnitude of the odds against such a claim. And yet that is exactly the likelihood of the claim presented by the negation of the principle of induction. And so, however unlikely that claim may be, its mathematical inverse is the likelihood of the principle of induction. Such a likelihood is so high that it's well beyond any connotations implied by the adjective "astronomical."

3e. Any negation of the inductive principle is arbitrary.

If I note that out of a theoretically unlimited set of competing propositions, one of them is produced by a generalized method, while all of the others could only be supported by special pleading, then I should favor the first method (and thus the proposition which it produces).

This isn't necessarily the case when comparing a set of propositions whereby some of them could be produced by one general method and others by another general method. This assertion only applies when it is the case that only one out of a given set of propositions can be produced by a method more general than a prejudicial "proposition X is superior" method. In that case, that first proposition should be treated as superior.

What I'm getting at in the above is the virtue of being non-arbitrary. Any proposition which can be produced by a general rule can be said to be non-arbitrary. Any proposition which could not be produced by a general rule is arbitrary. Non-arbitrary propositions are better than arbitrary propositions. This is because no given two competing arbitrary propositions have any rational basis upon which one could determine which should be adopted. If one of them had such a basis, it would constitute a general rule, and therefore make that proposition non-arbitrary. Out of any number of competing propositions, the one that can give you a rational reason for agreeing with it other than "I'm the best because I'm me" should be favored.

Otherwise, there is absolutely no way to narrow down agreement from among any two or more competing positions. Arguments could never be settled; conclusions could never be reached. If we don't value being non-arbitrary, then there can never be a basis to rule fairly between any given set of competing propositions. In fact, providing such a rule is exactly what the non-arbitrary proposition does; that's what makes it non-arbitrary.

As noted previously, there are a nigh-infinite number of principles I could follow other than that of induction. None of them can claim to have any greater chance than the others (including the principle of induction) of resulting in favorable outcomes. There are furthermore a nigh-infinite number of random possible universes. None of them can claim to have any greater claim to the others to being the real universe.

On the other hand, A) induction leads me to a specific course of action, whereas no other principle can lead me to anything other than twitching and drooling; and B) induction leads me to build a picture of one specific universe, whereas all others lead to merely random "white noise" outcomes, with all argument between them being which random result is in any way better than the infinite others, none of them providing criteria for "better," and there being no way to judge between those. None of the non-inductive pictures of the universe can explain to me how they constitute "signal" rather than noise. If they could, then they would be inductive.

To prefer a proposition supported by a general method over some number of competing propositions supported by no general method is the very basis of reason itself. It's the ancient-world judge holding court over a land-claim between four people, one of which holds a deed (or a claim to older law, or any claim that could constitute a "reason" whether good or bad), and the other three claiming that they deserve to win "just because." This picture amounts to the very basis of the word "reason."

Out of all of the possible descriptions of the universe, induction leads in the direction of one (or at least a very few). It provides a reason for that description. It is non-arbitrary. There are a theoretically infinite number of alternative ways to describe the universe, as many as there are random possibilities. We have not enumerated these because being relatively arbitrary, they are also meaningless.

Only a universe described by the principle of induction can be called meaningful or rational. Only a principle of behavior which responds to the principle of induction in kind can be called useful or reasonable.

3f. Regarding what induction does and does not necessarily entail.

As I mentioned much earlier on, I need not observe a crystal-clear perfect 100% understandable universe in order to call upon the principle of induction. All I require is that some consistency occur in my world. To the degree that the universe presents itself to me as consistent, that becomes my "known universe." That there is more that I do not understand is not a problem, because I have no way of measuring how much I can understand against how much I don't. The fact of the matter is, I don't know how much I don't know. As such, any ratio of knowable vs. unknowable universe is impossible to determine. As such, all I have to work with is the finite quantity of knowable universe that I've discovered, and I'll leave the rest to the side. The knowable universe is made up of whatever consistencies I've observed in my experience. But from even such a finite (and admittedly limited) measurement, I can extrapolate all of the principles outlined above.

Induction need not be about past results and future predictions. In fact, as a concept, it should not. In the above descriptions, I have casually spoken as if there were such a thing as time. But I make no such assumption (that there is a such thing as time) at this point. When all we've been talking about is the relationship between memory and expectation, how can I claim to have said anything if I didn't assume that time exists? Because I also haven't assumed that there is any real distinction between recollection from memory and current sensory experience. As I mentioned in the previous essay, memory and sensory experience are both treated as empirical input. They are merely that which is presented by my empirical environment ("This") to my subjective state of being ("I").

Thus, to watch two like events occur simultaneously with my senses is no different than comparing one event in my sensory environment to another in my memory. In either case, all I'm doing is noting that the situations and results are the same, and in either case the number of compared events is 2. In one version I see two simultaneous events in my sensory input. In another version I see one event in my sensory input, and another in my recollection from memory, and compare them simultaneously. The distinction between kinds of experience is not an assertion that I have yet made. Either way, I am merely comparing two experiential events (recollection vs. sense) in the here and now.

Induction isn't necessarily about time. It's about the expectation of a continued pattern (which might be in space or time).

Let's say that I spy into a room through a keyhole. I notice that the wall opposite the keyhole has a consistent wallpaper pattern on it. I assume that the entire wall is covered in the pattern that I have seen. I might be false in that assumption. In fact, it's quite possible that there's a painting hanging somewhere on that wall that would cover up the pattern in that spot. So if I predicted that that spot had the same pattern, I might be wrong. Nevertheless, it remains the case that I'm probably right, for any given square foot of prediction. Hanged paintings are less prevalent (on a square-foot basis) than wallpaper.

What this example does for space, we can also apply to time and expectation. That is to say, in the above example, expecting the wallpaper pattern to repeat in space is the equivalent of expecting past trends to continue in the future. But the above description makes a few assumptions regarding walls, wallpapers and hung paintings. It also doesn't properly describe the odds we deal with in real life. So let's try this:

I walk into a room, eyes shut, with my hand outstretched until I touch a wall. I place my palm firmly over where it landed on the wall, and open my eyes. I observe that the wallpaper is an evenly detailed pattern throughout, everywhere that I can see. The only place I can't see is under my hand (a tiny unobserved portion). What should I expect the odds to be that the portion under my hand is different than there rest of the pattern? Put another way, what should I expect the odds to be that my hand landed on the one small portion of the wall that was different from all the rest, if any was?

This better describes the likelihoods that we're looking at when we consider the problem of induction.

Some of the wallpaper pattern is in my memory. Some of it is in my senses. The challenge to the principle of induction is why I should expect the pattern to repeat beyond the area that I've already observed. Ultimately, my answer boils down to "what else ya got?" I've got something to go on. It presents itself to me as useful, likely, and non-arbitrary. Without a similar consistent-pattern view, nothing else you could suggest could possibly do the same. Any other possibility A) can provide no reason for why it should be useful to me, B) is incredibly unlikely, and C) is destined to be arbitrary, when compared to the entire range of possibilities.

Although there are many more formal and specific working definitions for induction, ultimately it amounts to "responding to the universe in kind." That is, whatever consistencies the universe demonstrates to us we take at face value, and behave as if those were guiding principles for our response. We do this because we can do no other; there is no other principle that can compete with it for our adoption.

We do not place belief in induction because some rationalist system has assigned it the value of "true." Rather, we do so based on its merits of usefulness, likelihood, and being non-arbitrary. Because of these merits, we align ourselves with the principle of induction, which is to say that we place belief in it. It is only from this principle that we have any chance of further establishing the kinds of conceptual frameworks will which lead us to the concept of truth. Induction is not to be adopted because it has the virtue of being true; rather, induction supports truth, which is not a virtue in and of itself. The merits of both induction and truth are: being useful, likely, and non-arbitrary.

Next up: I discuss conceptual frameworks and the development of truth.