originally by Michael Lensi, April 2022
Crater on earth’s moon named after David Hume.
Despite Kant’s brilliance, David Hume’s crux metaphysicorum is still very much with us today. According to the SEP (Henderson, L., “The Problem of Induction”, The Stanford Encyclopedia of Philosophy [Spring 2020 Edition], Edward N. Zalta [ed.], URL = <https://plato.stanford.edu/entries/induction-problem/>., hereafter “SEP”), “A number of philosophers have attempted solutions to the problem [of induction], but a significant number have embraced [Hume’s] conclusion that it is insoluble.” I would like to attempt a new solution here.
It is acknowledged by every early modern philosopher that we make observations of the world. That these observations count as knowledge is the hallmark of the empiricists. If we are to ever speak in anything but the past tense when it comes to factual knowledge, then, we must be able to convert these observations into knowledge, not necessarily of, but for the future. When an observation is thus accorded the status of a cause leading to an effect, such a conversion of knowledge from past to future is effected. That is a single prediction. When we expect we can repeat a prediction, we (implicitly or explicitly) posit a uniformity to nature.
Can we justify this process? At what point in the chain shall we justify it? Kant attacked this chain at about its earliest point by providing a priori causality as a necessary condition of experience. Arguably this secures everything downstream in the chain. My current endeavor is not to argue for or against this solution, but to re-inspect the chain and ask, is this really how empirical knowledge works in practice? If not, what is it, and how can we then justify it?
Working backward in the chain, I’ll first claim that although we do require nature to be uniform, what we mean in practice is that we may expect but do not require that observation is uniform. In fact, uniformity of observation - always observing the prediction or effect we expect - would imply that nature is “solved” by knowledge; epistemology and metaphysics would collapse into one. To expect the uniformity we require of nature to present itself in our observations of the world is to assert the end of philosophy.
Empirical knowledge, then, is the flexibility to accept and adapt in the presence of errant observation. All empirical work, in this view, begins with recognizing not uniformity, but the absence of uniformity. Induction, then, is merely the recognition that some observation has failed in some way. If that is a “downgrade” of induction from its heights as a “uniformity principle”, the upside to this framing of empirical knowledge and induction is that, as I’ll show, we can now justify induction on purely analytic grounds. Another advantage to this framework (over and above Kant’s - which is not up for dispute here) is that it would apply to natural objects in themselves, not just as they appear to us. I’ll not demonstrate this last claim. Here, I attempt only a new solution to the problem of induction - the crux metaphysicorum.
What Uniformity?
The first example of matter-of-fact knowledge given by Hume in the Enquiry is the astronomical phenomenon of sunrise (Enquiry, Section IV, Part I). Astronomical events are characteristically cyclical (to some degree). This means that modeling them to within a certain accuracy is trivial - being cyclical, the data itself is a good model. This is how the Babylonians were able to predict astronomical events before Newton, before Kepler, before Copernicus, before Ptolemy - without any further modeling than taking exquisite data. In other words, when astronomical events are the exemplar, it’s easy to conflate modeling and mere observation of the world. That is, it is easy to confuse the epistemological project of modeling-and-predicting the world with the way the world is.
The real epistemological project here begins, roughly, with Ptolemy, who noted discrepancies between new observations and the old observations. To solve this, he extended the concept of the cycle by nesting further cycles within the original, i.e. epicycles. Further modeling and explanation pivoted on further noted discrepancies, leading to Copernicus’ monumental shift of reference frames, then Kepler’s elliptical models, then Newton’s “universal” explanation of these, then Einstein’s “corrections” via the remodeling of space and time itself. At each stage, new modeling - new knowledge - begins with discrepancy between predictions of the old models and observation. In other words, the process of explanation (abduction) begins with and is grounded in induction which is nothing other than an observed discrepancy and the recourse that something in nature caused it, other than the cause we thought it was.
What I’m attempting to demonstrate here is that empirical knowledge does indeed rely on a uniformity to nature. But it is not an epistemological uniformity. Epistemological uniformity is the stagnation of knowledge. Epistemological non-uniformity is required. This is nothing other than recognizing that empirical, or Humean matter-of-fact, knowledge is never guaranteed; it is never uniform. Epistemologically, what is uniform is the use of a given model. But using a model is not knowledge. Building the model (abduction) is knowledge, and it begins in induction.
When Hume asserts and hands down to us his so-called uniformity principle, he has not yet teased this out. The uniformity principle is stated variously throughout his works but here it is from the Enquiry, Section IV, Part II, “From causes which appear similar, we expect similar effects”, or as characterized by the SEP:
Uniformity Principle (Humean), i.e. UP:
All observed instances of A have been B.
The next instance of A will be B.
As should be obvious by now, this can not serve as an epistemological statement or a valid framework for induction, because that is only the uniform employment of a given model. To do empirical work, induction can not be uniform. Like the use of astronomical data before more sophisticated modeling, we still need to separate out here what is actually uniform and what is not. We need to untie the metaphysical statement about the way the world is, and the epistemological statement of induction.
Untying the Knot
The uniformity we require is a metaphysical one that captures the way the world is. It may seem like a subtle reworking of the Humean principle but it is profoundly different:
Uniformity Principle (Metaphysical), i.e. UP-M:
All instances of A have been B.
The next instance of A will be B.
This statement is completely severed from observation and epistemology, as it should be. We might pause to note here that this is a trivial, definitional (relations-of-ideas a la Hume) claim, but I’ll demonstrate that later. Right now, we still need the corresponding epistemological claim, so let’s proceed thus.
Induction, by definition, is always uncertain, and never guaranteed in the way the metaphysical claim is guaranteed. Thus, to make the proper epistemological claim, consider how inductive knowledge is garnered in practice:
We observe facts. We [possibly] create or assume a model. We observe more facts. If conflicting with previous experience [or the model] (i.e. UP-M does not seem to hold), we do not discard the UP-M, we update or create a new understanding [model] of what happened. So that:
Uniformity Principle (Epistemic), i.e. UP-E:
All observed instances of A have been B.
The next observed instance of A will be B.
Unless what appears to be A happens to be an observation of A*.
In which case B* may occur instead of B.
(otherwise the non-occurrence of B is a contradiction of our metaphysical claim, UP-M)
We note that although nothing about the UP-E is uniform, we retain the nomenclature anyway, for ease of reference to its Humean origin. With this reworking of the uniformity principle, we can now finally justify induction on argumentative grounds. To do this properly, I will follow Hume’s original argument, in its reconstructed form per the SEP.
Solution to the Problem of Induction
Table 1: Hume’s Original Problem of Induction, Reconstructed (SEP §2)
Reading through the original argument in Table 1 above, we can see that the first horn fails because Hume’s original uniformity principle (UP) can not be demonstratively contradicted, and subsequently the second horn fails because a probable argument involving the UP is circular.
Replacing the UP with either the UP-M or the UP-E instantly resolves the argument at P4. That is, the negation of the UP-M is a contradiction, and the negation of the UP-E is a contradiction.
Let’s first work through why the negation of the UP-E is a contradiction. The negation of the UP-E actually leads to the negation of the UP-M. Let’s start with a straight demonstration of the UP-E, then its negation, using the following example (billiard balls, which Hume would appreciate):
A - blue ball rolls freely into unobstructed red ball at rest
B - red ball rolls away
A* - blue ball rolls freely into obstructed red ball (e.g. glued to table)
B* - red ball remains at rest
Starting with observation A and following the UP-E, we expect inference B. If we find that in (ostensibly) repeating A we do not get B, but instead observe B*, then it is not that A will not always give B (it will), but that what happened was we mistook A* for A. The UP-E (and the UP-E only) properly encodes the fact that induction is never certain.
The negation of this example would give not B* from A*, but B* from A. This is a metaphysical impossibility, i.e. it negates the UP-M. I had stated previously that the UP-M is a trivial, relations-of-ideas claim (and therefore its negation is a contradiction), and now it’s time to demonstrate this fact.
The definitional nature of the UP-M relies on its specificity, and uniformity, which is appropriate for a metaphysical claim, i.e. a fact about the world (not our knowledge of said fact, but simply the way it is in itself). Returning to the billiard ball example, in either case we are claiming nothing more than:
All instances of (A) billiard ball 1 colliding with billiard ball 2 under specific circumstances C have been (B) billiard ball 2 moving away with particular motion M.
The next instance of (A) billiard ball 1 colliding with billiard ball 2 under specific circumstances C will be (B) billiard ball 2 moving away with particular motion M.
In going from “all instances” to “the next instance” here, all we’ve done is leverage a definition. “The next instance” claim always follows from the “all instances” claim. In order for “the next instance” to ever not follow from the “all instances” claim would require a modification to a particular element of the claim, i.e. billiard balls which are not 1 or 2, circumstances which are not C, or motion which is not M. In which case, we have B* and/or A* as per UP-E.
Let me emphasize, as a definitional statement, the UP-M does not give us a scientific model. Induction and the UP-M set the stage for scientific modeling or any abductive explanations at all. Scientific models cannot be contradicted in the way the UP-M can; that is, they are not necessary. Induction is the flexibility to accept and adapt in the presence of errant observation. That is knowledge in itself (something has gone wrong with our old way of thinking!), and here I have shown it has a solid analytical footing. All we are saying at that point is that some explanation exists between some cause and its effect. We are not even required to give one. When we do feel compelled to give an explanation, the UP-M/UP-E will even hold under really bad explanations. This is why for so long in human history untenable explanations were proffered for inexplicable observations. Mistaking this further process of modeling for mere recourse to explanation is where we go wrong when we substitute methods appropriate for modeling (Bayesianism, Inference to the Best Explanation, Reichenbach’s Pragmatic Vindication, etc.) for justifications of induction.
At this point, we’ve shown that the negation of the UP-M is a contradiction (as a relations-of-ideas claim). We’ve shown that the negation of the UP-E leads to the negation of the UP-M. Thus the problem of induction fails at P4. QED. The problem of induction is no problem at all, and we have now set a proper foundation under inductive knowledge.
Conclusion
Early attempts at natural philosophy and inductive knowledge, taking a large cue from astronomical observation, led to a conflation of data (the way the world is) with models of that data (the heavens repeat themselves). Real epistemological work begins when one recognizes that one’s beliefs or models are conflicting with observation. Thus induction begins with non-uniformities; any expected uniformity comes from employment of a model and not actual expectations of the world itself.
Hume’s account of inductive knowledge as "From causes which appear similar, we expect similar effects.” (i.e. the uniformity principle, “UP”) is not the correct account or framing of inductive knowledge. Whereas Hume expects similar effects, I have shown that we do not expect similar effects. My claim is that inductive knowledge posits either exactly similar effects from exactly similar causes (a trivial metaphysical statement, “UP-M”), or it merely expects some cause is available to explain observed effects in any way different from previous effects (“UP-E”). That is the correct framework for inductive knowledge. Does Hume’s skeptical attack (his argument from habit) on the foundation of this knowledge hold up on this new account? No. In every case, once we associate a cause with an effect, we refer to the UP-M, the negation of which is a contradiction. This accounting holds up under a rigorous reconstruction of Hume’s argument.