Sunday, July 15, 2012

Simplicity as Evidence of Truth: How Do We Know It?

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This part 3 of a series on simplicity being evidence of truth.
  1. Simplicity as Evidence of Truth: Justifying Ockham’s Razor
  2. Simplicity as Evidence of Truth: Theories Tying Into Background Knowledge
  3. Simplicity as Evidence of Truth: How Do We Know It?
In this section I’ll have some concluding remarks to this series, one of which is this: how do we know that simplicity is evidence of truth? Whereas in part 1 of simplicity as evidence of truth I offered some justification for thinking that simplicity is evidence of truth, in this entry I’ll say a bit about how in practice we come to believe simplicity is such a guide for truth.

Recap

In part 1 of simplicity as evidence of truth I gave the following illustration to evoke the intuition that, ceteris paribus, the simplest explanation is the one most likely to be true. Suppose we investigate a new area of scientific research for which we have no background knowledge to tell us which theory is more probable and we study two variables: x and y. We have the following data:

Data Set 1
 
x0123456
y024681012


Let’s call the above situation the “Data Set 1 Scenario.” An equation presents itself for predicting the other values of x and y: y = 2x (we’ll call this equation 1), but that isn’t the only formula that fits the data. As Swinburne points out, all formulas of the following form (which we’ll call equation 2) yield the same data as well.[1]

Equation 2
 
y = 2x + (x − 1)(x − 2)(x − 3)(x − 4)(x − 5)(x − 6)z


Where z can be any constant or function of x. Although they agree with the data, the two equations may make very different predictions of unobserved data. For example, we can let z be x720 to get the following equation that predicts unobserved data differently from equation 1:

Equation 3
 
y = 2x + (x − 1)(x − 2)(x − 3)(x − 4)(x − 5)(x − 6)(x720)


Equation 1 predicts that when x = 9, y will be 18. Equation 3 predicts that when x = 9, y will be 270. If we were forced to go with either equation 1 or equation 3 to predict further data, which one would we choose? Obviously equation 1, and the reason seems clear: simplicity. There are literally infinitely many equations fitting Data Set 1 yielding infinitely many different y values for x = 9 (if nothing else, there are infinitely many numbers one could use for z), yet if one were forced to correctly predict what y would be for x = 9 and the consequences were sufficiently dire for predicting incorrectly (say, upon pain of ignominious death that one strongly wants to avoid), we would think it quite irrational to give any answer other than y = 18 for x = 9. This fact (if the stakes were sufficiently high it would be irrational to go with anything other than y = 18 for x = 9) suggests that simplicity is indeed a tool of rationality; we use simplicity as a guide for obtaining the truth.

Past Success Justification

The law of parsimony has been widely used in the history of science. How is it that we know simplicity is evidence of truth? One possible response is, “In science’s history, we’ve found that simpler theories tend to be better predictors.” One could say that the reason we believe simplicity is a guide for truth is through experience. Simplicity has served us well in the past, so probably it will serve us well in the future.

There are some problems with that “past success justification” approach however. Swinburne says the “in the past, simpler scientific theories have been the better predictors” claim is doubtful, since in science simpler physical laws have on many occasions been supplanted by more complicated laws.[2] Swinburne also adds:
But even if simplest theories have usually proved better predictors, this would not provide a justification for subsequent use of the criterion of simplicity, for the reason that the justification itself already relies on the criterion of simplicity. There are different ways of extrapolating from the corpus of past data about the relative success which was had by actual theories and which would have been had by possible theories of different kinds, if they had been formulated. “Usually simpler theories predict better than more complex theories” is one way. [3]
There are many ways to form a pattern of “theories that predict best” (or for that matter, true theories) that fit past experience (many of these giving different predictions about what future theories are likely to be better predictors), and surprise surpise, simplicity is big factor when choosing the “right” pattern. Astute readers may recognize the similarity between this situation and the Data Set 1 scenario, where there are innumerable theories that fit Data Set 1, predict that data set with perfect success, but give different future predictions.

If you’re skeptical of the existence of innumerable patterns fitting past experience, note that if nothing else one can construct an outrageously complicated disjunction of specific descriptions like “If (a man named Newton does such-and-such at such-and-such time) or (a man named Einstein does such-and-such at such-and-such time) or (...) or (...) ..., then the resulting equation will be at least approximately true.” If one objects saying, “Well, if Newton were named Smith, the same results would have obtained” one should note that (a) this still doesn’t change the fact that the massive disjunction perfectly fits past successes; (b) we can add “would have been” matters to the disjunction to get something like “If a man named Newton or Smith did such-and-such at such-and-such time)..., then the resulting equation will be at least approximately true.”

What’s more, just as equation 2 used a relatively complex set of parenthetical units (x − 1)(x − 2)…(x − 6) to help perfectly fit the observed data set, yielding infinitely many equations for infinitely many different possible future results (largely thanks to there being infinitely many choices for z), so too one can adopt a similar sort of approach for past successes where there are infinitely many models that perfectly fit past knowledge (and knowledge that would have obtained on other circumstances), give detailed predictions for the future, yet the models contradict each other over those future expectations. If nothing else, it remains true that there are many massive conjunctions (“If (A) or (B) or (C) or (D)…., then the theory is a good one”) that perfectly fit past successes yet give future different predictions of what is likely to be true in the future. Each of these infinitely many models, if it were used in the past, would have yielded great results. So clearly there are infinitely many models to fit past successes and at least some of these models (indeed, infinitely many) will contradict each other over future expectations. As noted in part 1 of this series, Swinburne notes that the criterion, “Choose the theory which postulates that the future resembles the past” is empty (infinitely many theories that are inconsistent with each other do that), and that to have real content we should change it to “Choose the theory which postulates that the future resembles the past in the simplest respect.” [4]

So how do we know that simplicity is a tool for rationality? In practice, this is something we know via a priori intuition. Even it is possible to construct a clever argument for why we should accept simplicity as a guide for truth, in practice that’s not why we do accept it. We accept it because it just seems to be true.

Not Just in Science

While a big focus in this series has been simplicity and science, the law of parsimony extends beyond science. Swinburne gives this nice illustration.
If we can explain an event as brought about by a person in virtue of powers of the same kind as other humans have, we do not postulate some novel power—we do not postulate that some person has a basic power of bending spoons at some distance away if we can explain the phenomenon of the spoons’ being bent by someone else bending them with his hands.[5]
Another illustration is the use of simplicity in the argument from ontological simplicity in my series on the moral argument.







[1] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 22.

[2] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 52

[3] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 52

[4] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 23

[5] Swinburne, Richard. Simplicity as Evidence of Truth (Milwaukee, Wisconsin: Marquette University Press, 1996), p. 59

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