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Here is an excerpt from the book An Introduction to Philosophy by Russ W. Payne (2015) which is causing me some confusion.

Mathematics had long served as the rationalist’s paradigm case of knowledge justified through reason alone. So we can make a powerful case for Empiricism by showing that math is really just an extension of logic. It remains debatable whether Frege, Russell, and Whitehead succeeded in showing this, but their attempt, and especially the powerful new system of logic they developed in making this attempt, constituted a powerful blow against Rationalism and inspired a group of empirically minded philosophers and scientists in Vienna to employ the same logical tools in analyzing and clarifying philosophical issues in science. As we will see, their ambitions were even grander since they also argued that much of what was going on in philosophy at the time was literally meaningless.

This is mentioned in the process of explaining logical positivism in Chapter 6: Philosophy of Science.

What I don't understand is why logic is at odds with rationalism.

I am under the impression that logic exists to make sure our reasoning is correct. And as Rationalism is founded on the belief that reason is the main source of human knowledge, it appears as if Rationalism relies on logic to verify whether certain beliefs are rational. So why would new developments in logic constitute "a powerful blow against Rationalism"?

Any help in clearing up my confusion will be greatly appreciated.

Edit: Here is the previous paragraph:

Logical Positivism can be understood as Empiricism, heavily influenced by Hume, and supercharged with powerful new developments in symbolic logic. The system of logic that we now teach in college level symbolic logic courses (PHIL& 120 at BC) was developed just over a century ago in the work of Gotlob Frege, Bertrand Russell, and Albert North Whitehead for the purpose of better understanding the foundations of mathematics. In Principia Mathematica, Russell and Whitehead made a strong case for analyzing all of mathematics in terms of logic (together with set theory). According to the argument of Principia Mathematica, mathematical truths are not truths justified independent of experience by the light of reason alone. Rather they are derivable from logic and set theory alone. Merely logical truths are trivial in the sense that they tell us nothing about the nature of the world. Any sentence of the form ‘Either P or not P’, for instance, is a basic logical truth. But, like all merely logical truths, sentences having this form assert nothing about how the world is. Logic doesn’t constitute knowledge of the world, it is merely a tool for organizing knowledge and maintaining consistency.

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  • Did the previous paragraphs discuss Gödel's incompleteness theorems of mathematical logic. These theorems certainly struck a strong blow against rationalism by showing that the formal methods of Frege et al will always fall short.
    – Nick
    Apr 13 at 21:23
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    Rationalism claims that reason is a major source of human knowledge, but logic is not a source of any knowledge in the relevant sense. It is a system for converting the same information into different forms, conclusions simply restate what is already "contained" in the premises. In contrast, mathematics looks like a source of new knowledge "by reason alone" (at first glance). But if it is reducible to logic this is just an illusion, all it does is conversion, just like logic. Hence the blow to rationalism, the paradigmatic example of knowledge "by reason alone" goes away.
    – Conifold
    Apr 13 at 23:05
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    @Logikal I am well aware of what you're saying. But I'm afraid that doesn't answer my question about why it is portrayed as "dealing a blow to Rationalism". Conifold's comment has answered that question so I think I get it now. I still haven't read the other answers as they are quite long but I'll read them in a few hours.
    – Oussema
    Apr 14 at 11:46
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    @Oussema, I would like to add that part of Conifold's answer is old slogan & false. All deductive reasoning does not have information already KNOWN. We can reason from assumptions that are clearly unknown. Why people regurgitate that saying is beyond me. Deductive reasoning can lead to NEW INFORMATION that one might have overlooked or not been aware of previously. Rationalism is about deductive reasoning as a whole not just math. There are concepts of reason not taught in math. Math selected some techniques. Rationalist say deductive knowledge can be gained without sensory experience.
    – Logikal
    Apr 14 at 11:56
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    @Logikal Had I just wanted to ask about Rationalism and logic in general, I wouldn't have included an entire excerpt from the book I mentioned and then went on and added the entire section that preceded it. I thought it was clear that I was looking for an answer within the context of the book. I'll be more clear next time. Also, please stop it with the words in capital letters, they make your comments seem aggressive.
    – Oussema
    Apr 14 at 12:56
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This is a bit subtle. I'm guessing "case of knowledge justified through reason alone" is pointing to the Kantian conception of mathematics as synthetic a priori knowledge. Frege really wanted to be a Kantian and expressed great admiration for Kant, whom had come up with the distinction between synthetic a priori and analytic a priori knowledge. But at the same time, Frege thought Kant was wrong about arithmetic being synthetic a priori and instead argued that arithmetic is analytic a priori.

Basically Kant had argued that the way we grasped 2 + 3 = 5 is counting up some things, so the result is synthetic. (He wrote something to the effect that no matter how much you stare at the concept of two and at the concept of three separately, the concept of five just doesn't pop up.) An analytic result for Kant was something like saying that the sum is greater than the part; basically something that follows from definitions. Frege instead argued that arithmetic could be axiomatized so it could be exactly like that.

(There is actually a bit more to be said here, as Frege tried to defend Kant's idea that geometry was synthetic, even though non-Euclidean geometry had been invented/discovered by that time. Frege argued that non-Euclidean geometry was non-intuitive, unlike arithmetic. Why geometry and arithmetic feature in this debate? Those were basically the only branches of math known to Kant.)

Gilles' book "Frege, Dedeking, Peano, and the Foundations of Arithmetic" is a really good philosophical read on this.

I suppose the way Payne presents this is because he thinks (if not argues explicitly--I don't have the book on hand to check) is that Kant is considered a rationalist.

You are however quite right to be riled by that because rationalism is often contrasted with empiricism. A prime exponent of empiricism at the time was J.S. Mill, who argued that mathematics was ultimately an inductive affair; we discover mathematical truths by examining nature basically. (At the very least Mill argued that axiom are discovered that way. Mill also argued that all results in arithmetic are empirical since we can basically check them that way.) Frege was also utterly opposed to Mill's conception of mathematics; unlike his expressed admiration for Kant, Frege could barely contain his disdain toward Mill. Which may seem a bit odd given that there doesn't appear to be a lot of difference between what Kant and Mill said about arithmetic. Frege's main beef with Mill was the latter held that numbers always come up from some application (i.e. that 1+2=3 always meant something like one apple plus two apples equals thee apples) whereas Frege (like Kant basically) was a Platonist, holding that the numbers that we operate on in arithmetic are ideal concepts.

Contra Frege, Gilles argues that from a modern perspective it's obvious that Frege was wrong about (non-Euclidean) geometry and effectively that the Kantian vision of mathematics as synthetic a priori knowledge had been blown up by Frege's work on arithmetic and the discovery of non-Euclidean geometry. It seems to me it's this erosion/destruction of the synthetic-analytic distinction with respect to mathematics as what Payne phrases as "Logic doesn’t constitute knowledge of the world, it is merely a tool for organizing knowledge [...] Mathematics had long served as the rationalist’s paradigm case of knowledge justified through reason alone." Payne also says "Logical Positivism can be understood as Empiricism", although Frege would have probably disagreed with that bit (if he ever identified as a "positivist", which I don't think he did).

There's a SEP page that discusses "logical empiricism" and it basically picks up from where Frege left the matters:

Carnap had not only studied with Frege, but like many of the logical empiricists he had started out as a neo-Kantian as well. So especially in view of Russell’s relatively more successful attempt at reducing mathematics to logic, it was perhaps natural that Carnap would consider both mathematics and logic as analytic. [...] But from fairly early on there was widespread agreement among the logical empiricists that there was no synthetic a priori, and that logic and mathematics and perhaps much else that seemed impervious to empirical disconfirmation should be thought of as analytic. The point of drawing the analytic-synthetic distinction, then, is not to divide the body of scientific truths or to divide philosophy from science, but to show how to integrate them into a natural scientific whole. Along the way the distinction clarifies which inferences are to be taken as legitimate and which are not. If, as Carnap and Neurath were, you are impressed by Duhemian arguments to the effect that generally claims must be combined in order to test them, the analytic-synthetic distinction allows you to clarify which combinations of claims are testable.

As another book notes somewhat hagiographically towards Frege, I'd say, but also noting the substantial difference I mentioned earlier between Frege and positivists:

Among the first appropriators of Frege’s work were the logical positivists who dominated philosophy in the period between the 1920s and 1950. Although among Frege’s three original promoters, Russell and Wittgenstein have greater individual philosophical stature than Carnap, Carnap was part of a movement—logical positivism—that had the broadest influence among those who first made use of Frege’s work. [...]

The positivists inherited from Frege an interest in meaning as expressed in language. Both Frege and his positivist successors were concerned to understand language and meaning, because they saw such understanding as a new and promising route to understanding the nature of human knowledge. Frege and the positivists took human knowledge to be best exemplified by scientific knowledge. (As we shall see, however, they understood this point in very different ways.) This approach to philosophy through a consideration of the nature of human knowledge is, of course, traditional. It had dominated the subject from Descartes through Kant. Kant was a source of inspiration for both Frege and his positivist successors.

In broadest terms, Frege added two things to this tradition. He added a concentration on language in the expression of knowledge. And he added a recognition of the power of logic to illuminate the structure of language and its contribution to the expression of knowledge. Frege is responsible for establishing and developing modern logic—a huge achievement. [...]

Frege’s positivist successors used Frege’s innovations. Yet they did so against a background of philosophical attitudes that Frege did not share. They differed with him, in absolutely fundamental ways, about both meaning and knowledge.

Although the positivists took up Frege’s focus on language, they appended to it an ideology of exclusivism. The only meaning that might be of any cognitive value was, for them, scientific meaning. It is this ideology that led positivism to be aggressively deflationary about those aspects of philosophy and culture that it could not assimilate to a scientific paradigm.

[...] Whereas Frege’s successors saw themselves as overturning traditional philosophy, Frege saw himself as continuing a philosophical tradition. Whereas his positivist successors applied reductionistic or deflationary attitudes to nearly all philosophical problems, Frege confined his reductionism to the attempt to reduce the mathematics of number to logic. He shows no special inclination to hold that reduction is a good method in philosophy generally.

Likewise Russell influenced positivism, but it's been disputed whether he substantially subscribed to that view himself. (More could be said on this, but I'm a bit short of time now.)

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Logical positivism may not be the perfect label, actually according to Hempel who's one of its major figures during its heyday in last century, it should be more properly called logical empiricism as described here:

Hempel never embraced the term "logical positivism" as an accurate description of the Vienna Circle and Berlin Group, preferring to describe those philosophers – and himself – as "logical empiricists." He believed that the term "positivism," with its roots in Auguste Comte, invoked a materialist metaphysics that empiricists need not embrace.

Now once you accept this label, it may be easier for you to further understand why it strikes a blow to rationalism, since the real opposition to rationalism is empiricism, not logic at all. All sane philosophers of various factions leverage all kinds of logic just same as other scientific or technical fields, and most part of logic has already been absorbed into mathematics thus it's not a key yardstick to identify epistemic schools of rationalism vs empiricism. And the main reason why logical positivism is essentially a type of empiricism is because of its central verification principle

Verificationism, also known as the verification principle or the verifiability criterion of meaning, is the philosophical doctrine which maintains that only statements that are empirically verifiable (i.e. verifiable through the senses) are cognitively meaningful, or else they are truths of logic (tautologies).

So during the heydays of logical positivism, it seemed all meaningful philosophy can be reduced to formal languages via many kinds of logic (predicate, modal, fuzzy, paraconsistent, etc) thus becomes a branch of science where empirical verification is the only source and standard to have a final say for every provable statement. If any philosophical problem cannot be reduced into such a formal system, then it was deemed meaningless and discarded...

However, logical positivism gradually went out of favor after notable critiques from Nelson Goodman, Willard Van Orman Quine, Norwood Hanson, Karl Popper, Thomas Kuhn, J. L. Austin, Peter Strawson, Hilary Putnam, and Richard Rorty. Major weaknesses exposed by those critiques are its atomistic philosophy of science, its central verification principle, and its fact/value gap. For example, the verification principle was itself unverified... Renowned rationalist logician Kurt Gödel in 1944 criticized Russell's logical empiricism as here:

Russell's no-class theory is the root of the problem: Gödel believes that impredicativity is not "absurd", as it appears throughout mathematics. Russell's problem derives from his constructivistic standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions . . . a notion being a symbol . . . so that a separate object denoted by the symbol appears as a mere fiction". Indeed, Russell's "no class" theory, Gödel concludes: "is of great interest as one of the few examples, carried out in detail, of the tendency to eliminate assumptions about the existence of objects outside the "data" and to replace them by constructions on the basis of these data. The "data" are to understand in a relative sense here; i.e. in our case as logic without the assumption of the existence of classes and concepts. The result has been in this case essentially negative; i.e. the classes and concepts introduced in this way do not have all the properties required from their use in mathematics. . . . All this is only a verification of the view defended above that logic and mathematics (just as physics) are built up on axioms with a real content which cannot be explained away"

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The split between Rationalism and Empiricism (note the capital letters) goes back to the 17th century. Basically, it is a dispute about the proper foundation of knowledge:

  • Rationalism holds that knowledge is (in one of several ways) founded in our mind and in the human capacity to use reason.
  • Empiricism holds that knowledge must (in one form or another) be founded in the evidence of our senses and physical events of the world

Both camps use reason and logic within their analyses, but they begin at different starting points. A somewhat simplistic example of the difference would be how they would view the statement "A ripe apple is red". For an Empiricist, this is a statement about the external world, in which we can physically measure wavelengths of light, sugar content, firmness, juiciness, and the like to show that 'ripeness' and 'redness' go together. For a Rationalist, this is more a statement of our inner world, in which 'ripeness' is based in purely intuitive understanding of what does and does not taste good, and 'redness' is a mental concept separate from any physical measurements. For an Empiricist, we measure an object to show that it is red, and work inwards toward the mind; for a Rationalist, we start with the understanding that the object is red, and work outwards toward the world.

This dispute has often been vituperative, in large part because it carries the onus of religious disputes. Rationalists are generally tolerant of metaphysical and transcendental concepts (souls, 'beingness', ideal forms, pure cognition...), while Empiricists generally abhor metaphysics as weakly religious maundering. That's why Empiricism is a mainstay in Anglophone philosophy (rooted in the rebellious protestantism of England and its colonies), while rationalism is pervasive in European philosophy (with its more traditional adherence to Catholic, Lutheran, and other organized, doctrinal faiths).

Of course, a statement like "A ripe apple is red" is both a statement about the external world and statement about our inner conceptual structures, and over the last hundred years or so — since Russell's exhaustive, brilliant, but ultimately failed effort to ground philosophy in Empiricism alone — the rift has been closing. Large segments of European philosophy have begun using empirical studies to back up their more expansive philosophical claims, and all but the most hard-bitten empiricists in the Anglophone world recognize the impact of language theory and critical analysis. I imagine the dispute will resolve itself, eventually: maybe in another hundred years..?

At any rate, any confusion should disappear with the understanding that 'Rationalism' as a movement isn't merely about using reason (which is something all philosophers do). It's an assertion that knowledge itself is based in reason, not in the world. If you like, think of the Rationalism/Empiricism divide as a disagreement about the ontology of knowledge — the real basis by which we say we 'know' — not about any more paltry concerns about how we use reason.

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  • Overall I agree with ur thesis, just want to express my nitpick skeptic about your near-term optimism about the unity of Hume's fork. From a said psychology experiment where the experimenter flash color "words" at people in various colors and ask the subjects what color the word 'was', they most likely tell you what color the word 'said', not the color they 'saw'; it concludes the invocation of the linguistic concept overrides the sensory perception entirely. So if this is true, seems most people naturally regard concepts dominant than senses, thus tend to assume they're more like rationalists Apr 14 at 23:39
  • ...another example is the famous historical debate btw rationalism and empiricism about the infinitesimal small dx devised by Leibniz. It seems infinitesimal small is beyond normal human being's sense experience, most students do calculus using it according to some rules simply by authoritative belief while find it's useful. It's actually just a clever new purely symbolic conceptual expression to count (measure) divisible space to try to resolve Zeno's paradox. So seems the dichotomy of empiricism and rationalism is an innate character of philosophy itself... Apr 14 at 23:48
  • @DoubleKnot: I admit, I'm an optimist. Doesn't mean I'm wrong, necessarily; all other things being equal I just prefer the cheerier side. Apr 15 at 3:02
  • then I admit I'm a minimalist, just staring at knowledg=justified (have to come from sense experience) true belief (a rational conception+will), I tend to believe these 2 are like dualistic un-annihilable pair innate in epistemology. Apr 15 at 3:10
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The above answers are all good and informative, but none of them seems to answer the question--at least not the question as I understand it. Here is how logicism helped the empricists:

Both rationalists and empiricists agree that deductive logic is a sound basis for reasoning about anything. Where they differ is in whether the human mind comes with a special faculty of intuition about nature that lets us just know things without having experienced them. The rationalist claim is that knowledge comes in three levels like this:

(1)        logic
             v
(2)      intuition
             v
(3)     experience

while the empiricists claim that only layers (1) and (3) exist, not layer (2).

The biggest argument in favor of layer (2) has always been based on mathematics. Mathematics doesn't seem to fit into either layer (1) or layer (3), so there must be a layer (2) to explain how we can just know things and those things apply to the empirical world.

Before Frege, the standard empiricist counter to this argument was to push mathematics into layer (3). There were various efforts to justify this, but none of them were convincing, so rationalists had the edge in this part of the argument.

What the logicists did was change their counter-argument. Now they would push mathematics into layer (1) instead of into layer (3). A lot of philosophers found this to be a more compelling response, so the main argument of the rationalists lost a lot of its force, and empiricism gained as a result.

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  • Empiricism gained as a result of claiming now that mathematics didn't require experience? Empiricism is the view that experience, especially of the senses, is the only source of knowledge. Contrary to your conclusion, your own answer suggests that logicism contradicts the empiricists' claim. May 8 at 9:05
  • @Speakpigeon, the empiricist position is that the results of logic are not genuine knowledge. May 10 at 7:36
  • I know that Wilbur and Wilfried are in the kitchen but I don't know that Wilbur is in the kitchen? I know that 1 + 1 = 2, but inferring from this that 1 + 1 = 2 is not genuine knowledge? I think that the empiricist claim is that logical reasoning doesn't produce new knowledge. However, I'm pretty sure that accept any old knowledge even if it is recycled by means of some logical reasoning. Empiricism is the view that experience is the only source of knowledge. Drinking tap water is still drinking water, I think. May 10 at 8:36
  • So the new counter-argument from logicism that mathematics is all inside logic and not at all inside experience is in fact conceding the point, unless we want to say that, because of that, mathematics is not knowledge. May 10 at 8:41

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