The great thing about math is that it’s certain. If something is mathematically true, you can be absolutely sure of it, because a mathematical proof is unarguably correct.
The set of rules for mathematical proofs is predicate logic, which guarantees absolute truth, so—some rationalists believe—they are the essence of rationality.
In the first half of the twentieth century, logical positivism attempted to marry predicate logic with scientific empiricism—which means generalizing from specific experimental data to universal truths.1 Proponents expected to create a complete and unassailable proof that this theory of rationality was correct.
Logical positivism conclusively failed around 1960, after multiple unfixable classes of trouble appeared. It was, however, the last serious attempt to build a general-purpose rationalism.
By “serious,” I mean that there was no known reason the project should fail when it began. In fact, there was every reason for excitement that it would succeed. Since its failure, no serious version of rationalism has been attempted, because no one has found plausible ways of dealing with the problems logical positivism encountered.
Logicism and probabilism
Epistemology traditionally distinguished “rationality” from “empiricism.” Rationality—correct thinking—derived new knowledge by deduction from existing knowledge, or from intuition. Empiricism derived new knowledge from sensory experience.
Rationalism and empiricism were opposing theories, and different epistemologists advocated one or the other. But by the late 1800s, it became clear that knowledge rests on reasoning and experience. In common usage, the word “rationality” now covers both. Logical positivism was one attempt to combine them.
On the other hand, “intuition,” which tradition considered an indispensable part of “rationality,” proved unreliable. People’s intuitions differ. When deductions differ, they can be made public and checked against each other. Intuitions are inherently private, so there is no way to resolve disagreements.
Part One of The Eggplant covers two major varieties of rationalism, which I’ll call logicism and probabilism.2 These are modern descendants of the rationalist and empiricist traditions, respectively.3 They are based on two mathematical systems, predicate logic and probability theory. Logical positivism took a logicist approach initially, but later incorporated probabilism as well.
Logic explains how you can deduce true sentences from other true sentences. From “All ravens are black” and “Huginn is a raven,” you can deduce “Huginn is black.” Classical logic (codified by Aristotle and mostly unchanged until the 1800s) had multiple well-known defects. The next several chapters cover various attempts to fix them by adding technical machinery. Some were successful to some degree, but overall the logicist project failed. Hardly anyone takes logic seriously as epistemology nowadays. Since formal logic is ancient history, it’s easy to ignore it, and many contemporary rationalists do.4
But it’s important to understand how and why logicism can’t work, because other rationalisms face the same problems, and because there are good reasons to think that no rationalist solutions are possible. Many errors of contemporary rationalism are due to ignorance of these issues.
The overall logical positivist plan:
- Apply rationality to itself. Use logic to prove that logic (as the correct account of rationality) works, and therefore is indeed rational. This “obviously must” be true, but we need to verify it, and then we’ll have an absolutely certain foundation before applying logic to other things.
- Use logic to prove that mathematics is correct. Eliminate the known problems that were bedeviling mathematics at the time, by clarifying definitions and inferences.
- Prove that the mathematical, scientific understanding of the world is correct, partly by proving that scientific empiricism is reliable.
- Fix all the squishy stuff like ethics and aesthetics and religion by reducing them to science.
Step 4 may have looked a bit iffy, but the others seemed like they should be straightforward. However, even step 1 failed.
The first difficulties the logical positivists ran into were inside logic itself, rather than in the attempt to connect it with empiricism. Some defects in logic were addressed, more-or-less adequately, with heroic effort.5 Eventually, Kurt Gödel and others proved mathematically that some defects cannot be fixed, even in principle. Logic is inherently somewhat broken, and there’s nothing that can be done about it.6
Specifically, what I said at the beginning of this chapter turned out to be false:
If something is mathematically true, you can be absolutely sure of it, because a mathematical proof is unarguably correct.
There are mathematical truths that can’t be proven.7 This was a major shock. The promise of rationalism was that, by rational methods, we can eventually come to know anything that is true. The modern worldview held that rationality’s power had no inherent bounds.
The discovery of several fundamental limits to what can be known—quantum indeterminacy was another—produced a crisis of confidence that eventually led to postmodernity. That is defined as “incredulity toward all grand narratives”; rationalism was central to most. Logic had been advertised as “the laws of thought”; so do its unfixable flaws mean that we cannot trust our own minds? Mid-century anti-rationalists gleefully seized on successive failures of rationalism to justify all sorts of harmful nonsense.
Step 2—proving mathematics correct—also failed. I won’t go into that, because the issues are highly technical. Also, mathematics mostly works, and it appears that its internal problems are irrelevant to its applications in science, engineering, economics, and so on.
The problem of induction
Logical positivists thought the problem of induction was the main difficulty in step 3. What rational procedure should we use to find universal truths? How much evidence about specific cases is adequate to verify a general conclusion? The difficulty is that it doesn’t matter how many black ravens you see, you can’t conclude “all ravens are black,” because maybe there’s a white one somewhere you haven’t seen.
Late in logical positivism’s evolution, after decades of failing attempts to find an absolute verification criterion, proponents reluctantly concluded that one could only have degrees of belief in universal truths; full confidence is impossible. Maybe, though, if you see more black ravens, you ought to be more confident they’re all black. Or at least that most are!
Probability theory was invented originally as a tool for winning at gambling. Some epistemologists developed it instead as a tool for understanding what “more confident” means. A great advance: theoretically, by emphasizing that you don’t simply believe that statements are true or false; and practically, because probability theory is uniquely effective in certain circumstances.
Logical positivists attempted to unify predicate logic with probability theory. That failed; even now it hasn’t been accomplished as mathematical system, much less as a philosophy of science. Nevertheless, probabilism replaced logicism as the dominant school of rationalism in the mid-twentieth century. Unfortunately, as an overall theory of belief, it is a non-starter. It has almost all the same problems as logic, plus others of its own, as we shall see.
Silence in the aftermath
By the 1960s, the obstacles to logical positivism seemed insuperable, and everyone gave up. Unfortunately, no one ever bothered to write a clear or detailed account of the reasons it failed, so every decade or two some movement reinvents it.
Here’s a 1976 interview with logical positivism’s former proponent A. J. Ayer:
MAGEE: Now logical positivism must have had actually some real defects. What do you now, in retrospect, think the main shortcomings of the movement were?
AYER: I suppose the greatest defect is that nearly all of it was false.
MAGEE: I think you need to say a little more about that.
AYER: Perhaps that’s being too harsh on it. I still want to say that it was true in spirit in a way, that the attitude was right. But if one goes for the details, first of all the verification principle [that is, the hypothetical correct way to do scientific induction] never got itself properly formulated. I tried several times and it always let in either too little or too much, and to this day it hasn’t received a properly logically precise formulation. Then, the reductionism just doesn’t work. You can’t reduce statements, even ordinary simple statements about cigarette cases and glasses and ashtrays, to statements about sense data, let alone more abstract statements of science.
Typical discussions of the movement’s collapse cover only the foundational crisis in logic and the failure to solve the problem of induction. That gives the hopeful impression that if these issues were overcome or bypassed, we could have a workable general epistemology. As for the first difficulty, we’ve learned that the internal mathematical problems are merely technical, with no practical consequences. As for the second, scientists increasingly adopted statistical software that promised reliable, automated answers to the problem of induction. “P<0.05! Take that, ivory-tower philosophical naysayers!”
However, we’ll see that logical positivism failed for several other reasons.8 These probably must afflict any other rationalism equally. Because these obstacles are less well-known, rationalists keep reinventing pentagonal wheels. (Logicism itself gets periodically rediscovered as “laws of thought” for artificial intelligence.)
A serious contemporary rationalism would have to acknowledge and address the difficulties explicitly.
- 1. “Positivism” is a vague word that in “logical positivism” means pretty much the same as “scientific empiricism.” The category “logical positivism” is also nebulous; different thinkers and works are included or not by different historians. I will use it in the broadest, vaguest sense possible: as rooted in Frege’s development of formal logic; as including “logical empiricism,” which is sometimes treated as a distinct movement; and as stretching into the 1950s work of, for example, Carnap and Reichenbach. It definitely doesn’t include logicism in artificial intelligence research, though, despite close similarities.
- 2. I am using the terms “logicism” and “probabilism” in ways that are reasonably mainstream but somewhat broader than is typical. “Logicism” in one narrower sense refers only to a position in the philosophy of mathematics. It is also used to refer to the logic-based research program in artificial intelligence.
- 3. There are other modern rationalisms, but logicism and probabilism are the most influential, and the others face most of the same problems.
- 4. Math students may learn some basics for use in proofs. Analytic philosophy students also get a bit because it’s supposed to help you think clearly, and it was important in the founding of their field. However, most universities no longer teach the subject seriously in either the math or philosophy department. It’s extremely cool stuff, even though it doesn’t work! Understanding logic does help you think clearly, although mostly not for the reasons it was supposed to. If a university course is not available, “Teach Yourself Logic” is a valuable guide.
- 5. Russell and Whitehead’s type theory is the central example. It was an extraordinarily complex and difficult fix for a technical problem that had no significant implications for the overall project.
- 6. I’m using the word “broken” informally. More precisely, there are several features you’d want and expect logic to have, but you can’t get all of them at once. There are many different systems of mathematical logic, each of which fails to provide at least one of the desired functions. It’s possible to prove that there isn’t some correct, better system that combines all of them.
- 7. “Some mathematical truths can’t be proven” is a somewhat impressionistic statement of Gödel’s First Incompleteness Theorem. The precise statement is highly technical and not needed here.
- 8. Some of these other problems were partly understood by some of the founders of logical positivism, notably Alfred North Whitehead and Ludwig Wittgenstein. They proposed alternative ways forward. Their new approaches acknowledged ontological nebulosity, worked with it, and pointed in meta-rational directions. Their accounts are difficult, vague, confused, and wrong in details, but they got some bits importantly right. Wittgenstein’s Philosophical Investigations is a seminal text of meta-rationalism. Ironically, histories describe it as one of the seminal texts of analytic philosophy, a rationalist school that has carefully ignored the book’s main insights. A fine explanation of that disjuncture is Ian Ground’s “The relentless honesty of Ludwig Wittgenstein,” The Times Literary Supplement, October 10, 2017.