1 Induction and the uniformity of nature

Nature is uniform. Not perfectly uniform, of course. But there are relatively time-stable objects with relatively time-stable properties, and there are detectable regularities between neighboring events.Footnote ¹ Nature has been uniform for all of human history, and—from what we can tell—for long before as well.

Does the observed past uniformity of nature provide us with any reason for thinking that uniformity will continue into the future? The most obvious argument that it does is viciously circular—it simply assumes that the past serves as a rule for the future, which is the very point at issue. We’ve known this since 1739, when Hume published the first volume of his Treatise of Human Nature.

Hume infamously used this point to challenge the good standing of inductive reasoning, as such. In effect, he argued that every inductive argument presupposes that nature is uniform across the observed and unobserved portions of reality. But because this uniformity claim is contingent, any successful argument for it must itself be inductive, and thus, it will presuppose the very uniformity claim at issue, making it viciously circular. If Hume was right, a successful justification of induction requires a noncircular argument that the observed uniformity of nature extends to the hitherto unobserved portions of reality.Footnote ²

There is one popular contemporary strategy for noncircularly establishing general uniformity on the basis of observed uniformity. This is the necessitarian strategy, which first uses inference to the best explanation (abduction) to infer the existence of nomic necessary connections in order to explain observed uniformity, then deduces the uniformity of the unobserved from these inferred necessary connections.Footnote ³ This strategy is controversial. Many philosophers doubt that strong nomic necessary connections can reasonably be inferred using abduction on observed uniformities.Footnote ⁴

Here I will explain an alternative noncircular strategy for vindicating belief in the uniformity of nature, one that does not appeal to either abduction or necessary connections. The strategy instead applies the widely discussed but controversial Copernican principle.

2 The Copernican principle and the delta-t method

With apologies to Aristarchus of Samos, Copernicus taught us that we are not observing the universe from a special spatial position. This is the general Copernican principle: ceteris paribus, your position as an observer is probably not special. This means that in certain circumstances and in a certain sense, you could just as well have been observing from a different position, and this is relevant to our reasoning. Copernicus applied this principle spatially, but it can also be applied temporally.

This basic idea leads to a completely general and widely applicable method—the delta-t method—for estimating the future duration of some event or process based solely on its past duration at the time of observation. This method was pioneered by, and has been widely applied by, the physicist J. Richard Gott.Footnote ⁵ It has subsequently been discussed very widely, even in popular books and articles.Footnote ⁶

Gott originally came up with the delta-t method when he visited the Berlin Wall in 1969. At the time, the wall was 8 years old. Gott used the past age of the wall, along with the Copernican principle, to estimate its future duration, reasoning as follows.

Suppose that the total time that the Berlin Wall will stand is T. The time between Gott’s observation, $t_{\text{now}}$ , and the beginning of the wall, $t_{\text{start}}$ , is 8 years. Gott reasoned that because $t_{\text{now}}$ was equally likely to be at any point in T, there was a $.5$ probability that $t_{\text{now}}$ occurred in the middle 50% of T:

$$\underbrace {\overbrace {\underline {\hskip 2pc} }^{25\% }|\overbrace {\underline {\hskip 5pc} }^{{\rm{50}}\% }|\overbrace {\underline {\hskip 2pc} }^{25\% }}_T$$

If $t_{\text{now}}$ is at the start of the middle 50% of T, as in this illustration:

$$\overbrace {{t_{{\rm{start}}}}\underline {\hskip 1pc} }^{25\% }|\overbrace {{t_{{\rm{now}}}}\underline {\hskip 6pc} {t_{{\rm{end}}}}}^{{\rm{75}}\% },$$

then:

$$T=4\times(t_{\text{now}}-t_{\text{start}}),$$

and:

$$(t_{\text{end}}-(t_{\text{now}}-t_{\text{start}}))=3\times8\;\rm{years}=24\;\rm{years}.$$

If $$t_{\text{now}}$$ is instead at the end of the middle 50% of T, as in this illustration:

$$\overbrace {{t_{{\rm{start}}}}\underline {\hskip 6pc} {t_{{\rm{now}}}}}^{{\rm{75}}\% }|\overbrace {\underline {\hskip 1pc} {t_{{\rm{end}}}}}^{25\% },$$

then:

$$T = {4 \over 3} \times ({t_{{\rm{now}}}} - {t_{{\rm{start}}}}),$$

and:

$$({t_{{\rm{end}}}} - ({t_{{\rm{now}}}} - {t_{{\rm{start}}}})) = {1 \over 3} \times 8\ {\rm{years}} = 2{2 \over 3}\ {\rm{years}}.$$

Putting these points together, we have that at $t_{\text{now}}$ (in 1969), the probability that the Berlin Wall will last between $$2{2 \over 3}\ {\rm{years}}$$ and $24\;\rm{years}$ beyond 1969 equals $.5$ :

$${\rm{Prob}}\left(2{2 \over 3}\ {\rm{years}} \lt ({t_{{\rm{end}}}} - {t_{{\rm{now}}}}) \lt 24\ {\rm{years}}\right) = .5.$$

Indeed, 20 years later, in 1989, the Berlin Wall fell. This successful prediction led Gott to publish his delta-t method.

The philosophical justification behind the delta-t method is rooted in the Copernican principle, applied temporally. Unless you have reasons for thinking that your time of observation is somehow special, you should reason as if you were randomly placed within the duration of some observed process or event.Footnote ⁷ This is the philosophical justification for the general delta-t method, which tells us, for a real number $0\leq r\leq1$ , there is probability r that

$$\left[\left({{1 - r} \over {1 + r}}\right)({t_{{\rm{now}}}} - {t_{{\rm{start}}}})\right] \lt [{t_{{\rm{end}}}} - ({t_{{\rm{now}}}} - {t_{{\rm{start}}}})] \lt \left[\left({{1 + r} \over {1 - r}}\right)({t_{{\rm{now}}}} - {t_{{\rm{start}}}})\right].$$

This general formula allows prediction, at credence level r, of how much longer a process will last after the time of observation based solely on the duration of the process at the time of observation.

The delta-t method is applicable only when you have no additional information about how long the observed process is likely to last. This means, for example, that we cannot apply the method to conclude that an 87-year-old man is more likely to live for a further 15 years than a 30-year-old woman. This is because we have empirical information about (for instance) the average human life span, relative life spans of human sexes, and the limits of human longevity. This kind of information would make the delta-t method inapplicable. So would information about a correlation between the probability of an observation and the duration of a process—when attending a wedding, you can’t apply delta-t to predict a quick divorce because in this case, it is certainly not as if your observation were randomly placed somewhere in the duration of the marriage.

Despite these limitations, Gott and others have applied the delta-t method very widely. Critics have usually argued that, at least for many of these applications, we do have additional information that makes the delta-t method inapplicable. Yet even these critics typically agree that when we have absolutely no prior information about the expected duration of a process, and there is no observation selection effect involved in our point of observation, the delta-t method is rationally applicable. I will return to these crucial background assumptions, but first let me explain how the delta-t method can be used to argue that the so-far-observed uniformity of nature will almost certainly continue into the future.

3 The Copernican principle and the uniformity of nature

Here is the argument. Let’s start with the provisional factual claim that humans have directly observed the uniformity of nature for the last $200,000\;\rm{years}$ . We did not observe the beginning of uniformity, but we know that it began earlier than $200,000\;\rm{years}$ ago. Later, I will discuss rolling back this assumption, but for now, let us proceed with it. Suppose that the total duration of the uniformity is U. We are at $u_{\text{now}}$ , and we know that $u_{\text{start}}$ was at least $200,000\;\rm{years}$ ago. That is, we know that:

$$(u_{\text{now}}-u_{\text{start}})\geq200,000\;\rm{years}.$$

To simplify things, let us assume:

$$(u_{\text{now}}-u_{\text{start}})=200,000\;\rm{years}.$$

Calculations based on this assumption will give us estimates based on the greatest lower bound for $(u_{\text{now}}-u_{\text{start}})$ . Now we apply the Copernican principle and the delta-t method at a $99.9\%$ confidence level:

$$\underbrace {\overbrace {{u_{{\rm{start}}}}\underline {\hskip 1pc} }^{0.05\% }|\overbrace {\underline {\hskip 8pc} }^{{\rm{99}}.{\rm{9}}\% }|\overbrace {\underline {\hskip 1pc} {u_{{\rm{end}}}}}^{0.05\% }}_U$$

Obviously, this illustration is not drawn to scale!

Assume first that our observation takes place at the earliest point in the $99.9\%$ confidence interval:

$$\overbrace {{u_{{\rm{start}}}}\underline {\hskip 1pc} }^{0.05\% }|\overbrace {{u_{{\rm{now}}}}\underline {\hskip 8pc} {u_{{\rm{end}}}}}^{{\rm{99}}.{\rm{95}}\% }$$

Then we have:

$$U=2,000\times(u_{\text{now}}-u_{\text{start}}),$$

and:

$$(u_{\text{end}}-(u_{\text{now}}-u_{\text{start}}))=1,999\times200,000\;\rm{years}=399.8\;\rm{million}\;\rm{years}.$$

If $u_{\text{now}}$ is instead at the other end of the middle $99.9\%$ of U:

$$\overbrace {{u_{{\rm{start}}}}\underline {\hskip 8pc} {u_{{\rm{now}}}}}^{{\rm{99}}.{\rm{95}}\% }|\overbrace {\underline {\hskip 1pc} {u_{{\rm{end}}}}}^{0.05\% },$$

then we have:

$$U = {{2,000} \over {1,999}} \times ({u_{{\rm{now}}}} - {u_{{\rm{start}}}}),$$

and:

$$({u_{{\rm{end}}}} - ({u_{{\rm{now}}}} - {u_{{\rm{start}}}})) = {1 \over {1,999}} \times 200,000\gt {\rm{years}} \approx 100{1 \over {20}}\ {\rm{years}}.$$

Putting these points together and assuming that the uniformity will end, the probability that the observed uniformity of nature will continue for between $$100{1 \over {20}}\ {\rm{years}}$$ and $399.8\;\rm{million}\;\rm{years}$ is $.999$ :

$${\rm{Prob}}\left(100{1 \over {20}}\ {\rm{years}} \lt ({u_{{\rm{end}}}} - ({u_{{\rm{now}}}} - {u_{{\rm{start}}}})) \lt 399.8\ {\rm{million}}\ {\rm{years}}\right) = .999.$$

Obviously, this entails that it is rational to be virtually certain that the uniformity of nature will continue through the end of all of our current experiments and endeavors:

$${\rm{Prob}}(({u_{{\rm{end}}}} - ({u_{{\rm{now}}}} - {u_{{\rm{start}}}})) \gt 1\ {\rm{year}}) \approx 1.$$

For smaller and smaller temporal intervals, we get closer and closer to rational certainty. Our expectation that the uniformity of nature will continue for the next few seconds is as close to rational certainty as empirical beliefs ever get, even if we use only the delta-t method applied to the observed duration of uniformity.

In full generality, suppose that some particular uniformity U has lasted for duration $t=t_{\text{now}}-t_{\text{start}}$ . What is the probability that U will last for another $${1 \over n}$$ th of t? To answer this, suppose you predict that U will last at least another $${t \over n}$$ . This prediction will only fail if the total duration of U, $T_{\text{total}}$ is less than $$t + {t \over n} = t(1 + {1 \over n}) = t({{n + 1} \over n})$$ . So, we have the following:

$${\rm{Prob}}\left(U\ {\rm{will}}\ {\rm{last}}\ {\rm{another}}{t \over {{n}}}\right) = {\rm{Prob}}\left({T_{{\rm{total}}}} \ge t\left({{n + 1} \over n}\right)\right).$$

To calculate this probability, we can use the Copernican principle to determine the probability that our observation point is in the last $(n+1)$ th of U. The odds against this are n to 1, so we have the following:

$${\rm{Prob}}\left(U\ {\rm{will}}\ {\rm{last}}\ {\rm{another}}{t \over {{n}}}\right) = {n \over {n + 1}}$$

And this holds for any n. So, the probability that U will last for another $${1 \over {100}}$$ th of t is $${{100} \over {101}}$$ , and for divisions of t into smaller intervals, the relevant probability gets closer and closer to 1.Footnote ⁸

Even if we knew nothing save that the uniformity of nature has lasted for over $200,000\;\rm{years}$ , it would be rational for us to be virtually certain that said uniformity will continue into the near future. On its face, this reasoning does not circularly use inductive reasoning, nor does it use abductive reasoning, nor does it infer the existence of controversial necessary connections.Footnote ⁹ Assuming this is right, we have a justification for induction that depends only on the Copernican principle applied temporally by way of the delta-t method. Given this, we can rationally infer that the observed uniformity of nature will almost certainly continue into the near future, and will quite probably continue into the distant future as well.

4 Trouble with the Copernican principle?

As far as I’m aware, nobody has yet attempted to apply the Copernican principle to the problem of induction. At least, not in anything like this form.Footnote ¹⁰ Perhaps this is because the general Copernican principle, understood as vindicating the delta-t method, is controversial.

For my purposes here, most of these controversies can be set aside. My overall argument is conditional: If the delta-t method can be applied to observed uniformity, then we have a noncircular reason for thinking that uniformity will continue into the future. Those skeptical of the method itself will resist discharging this conditional.

Yet, as noted earlier, even many severe critics of the delta-t method have admitted that it is applicable in certain special circumstances. Not every critic will agree, as some will reject the entire framework or argue that certain Bayesian assumptions preclude Gott-style reasoning.Footnote ¹¹ But I believe it is true even of Gott’s most hostile critics, such as Carlton M. Caves. Initially, Caves suggests that a proper Bayesian analysis shows that Copernican reasoning using Gott’s rule would be applicable only if we knew that a phenomenon was in progress but had no information about its past duration. But according to Caves, this blocks Gott-style applications because obtaining information about the past duration immediately makes the moment of observation special, thus undermining the Copernican approach and blocking any projections.Footnote ¹²

However, later in his discussion, Caves discusses a possible rejoinder to this objection, writing:

There remains disagreement about exactly what an “uninformative” prior is, in Bayesian terms, but even most of Gott’s critics admit that if our prior was uninformative, in the relevant sense, then Copernican reasoning would be applicable.Footnote ¹⁴ Yet this is—arguably, at least—exactly the situation we are in when faced with the uniformity of nature.

At least this seems to be the case when we face Hume’s problem in the internalist sense in which he originally posed it. Hume allows us our direct observations and past information about said observations, but little else. The allowance for past observations includes observations made by other humans, along with records of the past. If we were to instead allow only information derived from our own personal memories, then obviously the numbers would change, but the basic epistemic situation would not.

For this reason, I have not focused on worries that the base claim I started with—that humans have observed uniformity for 200,000 years—can only be arrived at with inductive reasoning and is thus, in this context, circular and therefore illegitimate. I think that there are some potentially promising responses to this sort of worry.^{Footnote 15,Footnote 16} But even if these responses don’t ultimately succeed, all hope is not lost for applying the Copernican principle to the problem of induction.

Even limiting our observations of uniformity to the uniformity that we have personally observed allows a reasonably long duration of “observed” uniformity—for adult humans, corresponding to the majority of our lives. Each of us can then use the Copernican strategy to get a reasonably high rational probability that uniformity will continue for the next small fraction of our lives as an observer. This pure version of the Copernican strategy is still available even if the just-discussed worries about the impure version prove decisive.

I won’t reproduce the details of the full calculation, but for even an average 20-year-old with 15 years of memories of observed uniformity, we have, using the same kinds of assumptions from earlier, the following:

$${\rm{Prob}}\left(2{3 \over 4}\ {\rm{days}} \lt ({u_{{\rm{end}}}} - ({u_{{\rm{now}}}} - {u_{{\rm{start}}}})) \lt 30,000\ {\rm{years}}\right) = .999.$$

Treating the left-hand side as a lower bound, we can see that in any given moment, the delta-t method sanctions high confidence approaching certainty that the observed uniformity will continue into the next day, let alone the next moment. So, even the “pure” version of the strategy makes significant progress.

If we were to allow only our observations in a given moment, things are obviously much different, but this is not a problem with the Copernican strategy—everyone will have difficulty justifying a story of the world starting from this extremely limited basis.Footnote ¹⁷

There are questions that can be raised about the Copernican argument. Yet many philosophers and scientists believe that delta-t reasoning is at least approximately applicable in cases where we observe some phenomenon without having any prior information about its expected duration or about positive or negative correlations of our observations with the duration of the phenomenon.Footnote ¹⁸ Even this is not beyond all dispute. Some critics are skeptical of all predictions made in the absence of specific empirical information.Footnote ¹⁹ I think that this would cripple our ability to reason, but I won’t try to refute the position here.Footnote ²⁰

Most of us think that Copernican reasoning is at least occasionally applicable. And that, if it is ever applicable, it is likely applicable to cases where we don’t have prior information about the duration of a process. That is exactly the position we find ourselves in when facing down the infamous Humean problem of induction.

To be clear: I am not simply making the claim that there is a method that, if applicable, solves the problem of induction. That kind of conditional is not always interesting. Magic would solve the problem of induction, if magic were applicable to the problem. The point here is that there is a plausible, non–ad hoc method for making predictions of duration absent any empirical information, making the method apparently tailor-made for Hume’s problem. Even critics of the method generally agree that the method is sometimes applicable, and if sometimes applicable, applicable in the situation in which it has been applied here.

Yet despite this, the method has never been applied to Hume’s problem. My modest goal here has simply been to bring it to the attention of philosophers that Copernicus just might be able to vanquish Hume.