The Problem of Induction

-Submitted by David Drumm (Nal), Guest Blogger

Induction is the type or reasoning from specific examples to general statements about reality. Induction is often used to justify the claims regarding the future. The problem of induction was first formulated by David Hume (left), though he didn’t use the term “induction.” The problem of induction occurs when we ask ourselves “how do we know that induction is a reliable rational tool?” The usual answer is that induction has worked in the past. But that answer justifies induction inductively, using circular reasoning (a logical fallacy).

How can we justify using induction to predict the future non-inductively?

Throughout our lives, but particularly during infancy, we learn about reality by perceiving and interacting with it. We form a mental model of reality. One aspect of reality that we learn is that many aspects of reality are not completely random with respect to time. A reality in which events happen randomly from one time interval to the next, would appear very different from the reality as we perceive it.

Reality is a set of random processes. Some of the random processes are completely random from one point in time to another point in time, for example, the flipping of a fair coin. The outcome of a flip at one time is independent of the outcome at another time. There is no correlation between from one point in time to another point in time. Other random processes do possess a high degree of correlation, for example, the rising of the sun from one day to the next. Other random process, like the weather, exhibit some short term (hourly) correlation but less correlation long term (weekly) and then more correlation (yearly). Random processes that display correlations that depends only on the difference between points in time are described as wide-sense stationary (WSS).

Reality is a set of WSS random processes, each with its own correlation. This is a mental model that has been created through experience. How do we know that in the future reality will still maintain its WSS property? We don’t. However, if reality ever lost its WSS property, the effect would be complete and utter chaos. With the loss of the WSS property, events that were previously improbable would become probable.

But aren’t we assuming that the WSS property is an inherent characteristic of reality? Aren’t we assuming the “uniformity of nature” will continue over time? Maybe, but we have shown that the absence of the WSS property would alter reality as we know it.

Using reality’s WSS property, we can predict, with varying degrees of confidence, future events. If there is no independence between present events and past events, for a particular random process, in the set of random process that constitutes reality, the probability that an event (A) will occur given that event (B) occurred, is strictly greater than the probability of event (A). That is, P(A|B) > P(A). Therefore, the probability that the sun will come up tomorrow given that it came up today is greater than the probability that the sun will come up tomorrow. The probability that the sun will come up tomorrow given that it came up today and yesterday is greater still. This can be continued over the millions of sunrises, but it will never equal one. There’s always the chance that a rogue black hole will happen by and ruin our day.

Christian apologists often use the problem of induction to claim that “Science is based on faith too, so there!” However, their justification for using induction, that God did it, is unsatisfying,

James Clerk Maxwell, formulator of classical electromagnetic theory, in 1850 wrote:

The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man’s mind.

H/T: Massimo Pigliucci, Eliezer Yudkowsky.

42 thoughts on “The Problem of Induction”

  1. “The intense view of these manifold contradictions and imperfections in human reason has so wrought upon me, and heated my brain, that I am ready to reject all belief and reasoning, and can look upon no opinion even as more probable or likely than another. Where am I, or what? From what causes do I derive my existence, and to what condition shall I return? Whose favour shall I court, and whose anger must I dread? What beings surround me? and on whom have, I any influence, or who have any influence on me? I am confounded with all these questions, and begin to fancy myself in the most deplorable condition imaginable, invironed with the deepest darkness, and utterly deprived of the use of every member and faculty. [Hume, Human Understanding]”

    Dont you just love that Hume.

  2. “tide goes in, tide goes out. never a miscommunication. you can’t explain that.”

  3. Buddha,

    I am sure that you have tried Bi-location…that is a St Germain topic if I have ever seen one…but the again…I want to know why…and here you all want to know how….I suppose, I’ll always be just one base behind or ahead…never know which…

  4. I can top that, AY.

    I can travel without moving.

    😉

  5. I am sure Buddha has tried driving with no hands….I am pretty sure of that….

  6. AY,

    I was thinking more along the lines of Shirley MacLaine than Jesus.

  7. The traditional wooden bridge had a life expectancy of only about nine or ten years at the most. It was discovered early on that covering a bridge structure could extend its life expectancy to eight or nine decades.

    I live in an area with several covered bridges and am only a short distance from the longest span covered bridge in the country, built in 1882 with a length of 154.3 feet. I know a lot about that particular bridge and have taken hundreds of pictures of it from every angle in every season. After 129 years, it is in great shape. Because of people crossing it in heavy SUVs like Suburbans and Hummers putting it at risk, it is now limited to foot and bicycle traffic.

    http://www.google.com/search?hl=en&safe=off&q=covered+bridge+elizabethton+tn&bav=on.2,or.r_gc.r_pw.&um=1&ie=UTF-8&tbm=isch&source=og&sa=N&tab=wi&biw=1280&bih=569

  8. AY,

    You’re talking about randy teenagers and those conducting clandestine affairs …

    I live in a area which has many covered bridges and that’s what all the tour guides and Park Rangers tell visitors … bridges were mainly constructed of wood which deteriorates with exposure to the elements. Covering the bridge protected the base which caused the bridge to last longer. That’s my story and I’m sticking to it.

    Buddha can walk on water and has no need of bridges … he did it yesterday, he did it today, so I’m going to bet he can do it tomorrow.

  9. Blouise,

    And if the GOP has anything to do with it…it is simulated concrete over stucco….but charge the concrete price…

  10. See Blouise,

    You are all wet….They Built the covered Bridges…to stay dry….then they could finish the remained….smart folks aren’t they….that is why….How they built it…Buddha can answer that…

  11. AY,

    Covered bridges last longer because wood deteriorates. Now, if one sees a covered bridge constructed of reinforced concrete one can rightly assume Congressional Earmarks.

  12. Buddha and AY,

    Peasants!

    My philosophy: When the rains come, if anyone, who doesn’t wish to be, is left without shelter then all the gods of our invention have failed.

    (I watch all reruns of NUMB3RS)

  13. Ah but Blouise…have you ever pondered why they built “Covered” Bridges…. That is one for thought….

  14. That is why “why” is never the interesting question, Blouise.

    “How” is the interesting question.

  15. And yet we experience rain, sunshine, seasonal changes, illness, birth and all of these require physical reactions on our part if we wish to live another day. The need for physical action is never questioned.

    We encounter problems only when we have time to sit and contemplate the why of it all. We invent gods … we invent philosophies … and then we argue over which one is more correct. Until the rain comes and we need to seek shelter.

  16. In other words “Warning: Past performance is not an indicator of future performance.”

    We assume the universe plays by the rules because we don’t know enough to catch it cheating.

  17. Karl Popper’s wikipedia page has an excellent section on the “problem of induction.” I’ll paste it here for all to read:

    “Among his contributions to philosophy is his attempt to answer the philosophical problem of induction as emphasized strongly by David Hume. The problem, in basic terms, can be understood by example: given that the sun has risen every day for as long as anyone can remember, what is the rational proof that it will rise tomorrow? How can one rationally prove that past events will continue to repeat in the future, just because they have repeated in the past?

    Popper claims to have found a solution to the problem of induction. His reply is characteristic, and ties in with his criterion of falsifiability. He states that while there is no way to prove that the sun will rise, it is possible to formulate the theory that every day the sun will rise—if it does not rise on some particular day, the theory will be falsified and will have to be replaced by a different one. Until that day, there is no need to reject the assumption that the theory is true. Neither is it rational according to Popper to instead make the more complex assumption that the sun will rise until a given day, but will stop doing so the day after, or similar statements with additional conditions.

    Such a theory would be true with higher probability, because it cannot be attacked so easily: To falsify the first one, it is sufficient to find that sun has stopped rising; to falsify the second one, one additionally needs the assumption that the given day has not yet been reached. Popper held that it is the least likely, or most easily falsifiable, or simplest theory (attributes which he identified as all the same thing) that explains known facts that one should rationally prefer. His opposition to positivism, which held that it is the theory most likely to be true that one should prefer, here becomes very apparent. It is impossible, Popper argues, to ensure a theory to be true (but not fatal, since even false theories may have true consequence); it is more important that they can be eliminated and corrected as easily as possible if false.

    Popper and Hume agreed that there is often a psychological belief that the sun will rise tomorrow, but both denied that there is logical justification for the supposition that it will, simply because it always has in the past. Popper writes:

    “I approached the problem of induction through Hume. Hume, I felt, was perfectly right in pointing out that induction cannot be logically justified.” (Conjectures and Refutations, p. 55)

    To Popper, who was an anti-justificationist, traditional philosophy is misled by the false principle of sufficient reason. He thinks that no assumption can ever be or needs ever to be justified, so a lack of justification is not a justification for doubt. Instead, theories should be tested and scrutinized. It is not the goal to bless theories with claims of certainty or justification, but to eliminate errors in them:

    “there are no such things as good positive reasons; nor do we need such things […] But [philosophers] obviously cannot quite bring [themselves] to believe that this is my opinion, let alone that it is right” (The Philosophy of Karl Popper, p. 1043)”

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