QuestionsCan we be certain of anything?There is certain knowledge that does not come from experience
Can we be certain of anything?Yes, through reason

There is certain knowledge that does not come from experience

Synthetic a priori judgements; Kant's Copernican revolution; 7 + 5 = 12

Immanuel Kant 1724 · 1804Set the limits of reason; act from duty.
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What it claims

There is synthetic a priori knowledge; it is possible because the objects of experience conform to the forms of our mind; and it holds only for phenomena.

Why

Mathematics is certain and teaches us something new: this is possible if objects conform to the forms of our mind.

The same idea through history
David Hume1748Hume distinguishes relations of ideas (certain, but telling nothing about the world) from matters of fact (informative, but without necessity). Kant wants to show a third kind of knowledge, which adds to our concepts and yet remains necessary.
Immanuel Kant1787Kant sums up his whole project in one question: "How are synthetic a priori judgements possible?" His answer: because the mind itself puts space, time and the categories into experience.
In short

"There is certain knowledge that does not come from experience" is an argument by Immanuel Kant (Kant, Critique of Pure Reason, 2nd ed., 1787 (introduction and preface), 1787) on the question "Can we be certain of anything?" (answer: yes, through reason). Its claim: There is synthetic a priori knowledge; it is possible because the objects of experience conform to the forms of our mind; and it holds only for phenomena. According to AskPhilosophy's logic check (Z3 solver), the conclusion does follow from its 5 premises. AskPhilosophy presents 3 objections, 2 of them with a documented reply, and the original text behind each premise, checked word for word.

Kant, Critique of Pure Reason, 2nd ed., 1787 (introduction and preface)
Valid

Does everything we know begin with experience? Kant answers: yes, but not everything comes from experience. When you add 7 and 5, you know for certain that the result is 12, always and everywhere, without needing to test it. Yet 12 is not hidden in the idea of "7 plus 5": you have to add something, by counting on your fingers for instance. Kant calls this kind of knowledge "synthetic a priori": it teaches something new (synthetic) and it is certain before any experience (a priori). He finds it in mathematics, in pure physics and in the principle "every change has a cause". How is it possible? His answer, which he compares to Copernicus's: it is not our knowledge that conforms to things, it is the objects of experience that conform to the forms of our mind (space, time, categories such as cause). The price is heavy: this certainty holds only for the world as it appears to us, not for things as they might be in themselves. The argument is presented here in its strongest form; premise P6 is the one Kant assumes without writing it down.

P1We have judgements that are necessary and strictly universal, with no possible exception: those of mathematics (7 + 5 = 12, the straight line is the shortest path) and the principle "every change must have a cause".
P2Experience teaches us that things are so, never that they could not be otherwise, and it gives only an assumed universality ("so far, no exception"). A necessary and strictly universal judgement therefore does not come from experience: it is a priori.
P3These same judgements are synthetic: their predicate is not contained in their subject. Analysing the concept of "7 + 5" does not give 12; analysing the concept of "straight line" does not give "the shortest". One must go beyond the concept, relying on intuition.
P4If our knowledge had to conform to objects, we could know nothing about them before meeting them, hence nothing a priori. Synthetic a priori knowledge is therefore possible only if the reverse holds: the objects of experience conform to the forms of our mind (space, time, categories).
P5If the objects of experience conform to the forms of our mind, then what we know a priori holds only for phenomena, that is, for things as they appear to us, and not for things in themselves.
P6 · implicit premiseA judgement that is both a priori and synthetic is real knowledge, and not a mere illusion of our mind or a habit: mathematics and pure physics are genuine sciences.
So there is synthetic a priori knowledge; it is possible because the objects of experience conform to the forms of our mind; and it holds only for phenomena.

click a premise to reject it

If you accept every premise, logic forces you to accept the conclusion. So the debate is about the premises.
Checked by a logic solver: the premises cannot all be true while the conclusion is false.
Without the implicit premise: Without P6, the computer finds a case where the judgements are a priori and synthetic but are not knowledge (for example a habit of the mind, as Hume says of causality): the conclusion fails.
With P6 the argument is valid: the judgements are a priori (P1, P2) and synthetic (P3), so by P6 they are synthetic a priori knowledge; P4 draws from this that objects conform to our mind, and P5 that this holds only for phenomena. Without P6 it no longer is: there could be necessary, synthetic judgements that are mere beliefs with no value as knowledge. P6 is what Hume (for causality), Reichenbach (for geometry) and Quine (for the very notion of "synthetic") each contest in their own way.
Analogy. Put on red-tinted glasses that you cannot take off. You know for certain, before looking at anything, that everything you will see will be red: this certainty does not come from what you have seen, it comes from your glasses. But you cannot know what colour things are without glasses. The image is an aid to explanation, not Kant's.

Objections

3

Each objection is a conversation: click to see who replies and where the debate stands.

David Hume (as cited by Kant) 1748 · attacks premise 1

The principle of causality is not necessary. We have only seen one event follow another very often, and habit makes us expect the second when the first occurs. This necessity is in our habits, not in things: no reasoning justifies it (see the argument on induction).

Kant replies
Kant 1787 · Kant's reply (1787)
Kant replies that the concept of cause contains the idea of a necessary connection and of a rule without exception. If it were derived only from habit, as Hume does, the concept would vanish: we would have a subjective expectation, not the idea of a cause. And he adds that without a priori rules, experience itself could not find its certainty.
The debate goes on (our summary) The disagreement concerns proof. Kant shows what the concept of cause is, and that we could not have ordered experience without it. A defender of Hume answers that this only proves that we necessarily think in terms of causes, not that the world really obeys causes. Kant half accepts this: causality holds for the world as it appears to us (P5), not for things in themselves.
H
Hans Reichenbach 1920 · attacks premise 3

The geometry that seemed to impose itself on the mind, Euclid's, is only one geometry among others (Lobachevsky, Riemann). And Einstein's theory of relativity describes real space with a non-Euclidean geometry. So the principles of space are not necessary and eternal truths of the mind. According to Reichenbach, the a priori in Kant's sense (true forever) must be given up, keeping only a "constitutive" a priori: principles that serve to build our picture of the world, but which experience may lead us to change.

no reply
Kant never answered this objection.
The debate goes on (our summary) The sensitive point is geometry: Kant's example is the straight line as the shortest path between two points, a principle of Euclid's. A Kantian may answer that other geometries are thinkable without being intuitable, but the debate remains open. Reichenbach himself does not reject Kant's whole idea: he keeps a part of it, a revisable one.
Willard Van Orman Quine 1951 · attacks premise 3

The boundary between analytic and synthetic judgements has never been drawn clearly: it is, says Quine, an article of faith of the empiricists. Without it, Kant's question (judgements that would be both synthetic and a priori) loses its footing. Quine adds that no statement is immune to revision: any one can be held true provided we adjust enough of the rest of our beliefs.

Kant replies
Paul Grice and Peter Strawson 1956 · Grice and Strawson's reply (1956)
In "In Defense of a Dogma", Grice and Strawson argue that the analytic/synthetic distinction is not worthless: it is used without difficulty by all speakers, and one cannot demand that a notion be defined in terms clearer than itself in order to make sense.
The debate goes on (our summary) Even if the distinction survives, Quine changed the debate: for him there is no domain of a priori truths sheltered from experience. Mathematics itself would be part of the web of our beliefs, revisable like the rest.