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.

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