As I was reading "The problems of philosophy" by Bertrand Russell I got the impression that Kant was right and Hume was wrong in the case of a priori knowledge of synthetic mathematic nature and I'm confused about that.

As I understand the analytic knowledge is known just by definition of the system we create so, therefore, a posteriori analytic knowledge is impossible.

I found the example which illustrates why mathematical knowledge is synthetic a prior. It goes like this: we cannot experience four dimension world and so we can assume it's a priori, but as I understand we just add another dimension by new variable, so it's still just known by the definition so it's still analytic knowledge.

Another example: "His stock instance was the proposition 7 + 5 = 12. He pointed out, quite truly, that 7 and 5 have to be put together to give 12: the idea of 12 is not contained in them, nor even in the idea of adding them together"

This part is important for me: "nor even in the idea of adding them together". For me in definition of adding them together contained in them!

Where is my mistake? Or maybe it's just a matter of your intuition and no one is right in this matter?

Also if it comes to analytic/synthetic knowledge, as I'm determinist not believing in the real randomness I don't believe if there is any synthetic knowledge. Is it valid to have such position?

  • 1
    Is your question why Kant was right (he arguably was not) or why/whether Russel's writing suggests that Kant was right?
    – Philip Klöcking
    Commented Oct 18, 2018 at 9:49
  • You're right, that was misleading Commented Oct 18, 2018 at 9:59
  • I think that the reading of "7 and 5 have to be put together to give 12: the idea of 12 is not contained in them, nor even in the idea of adding them together" is : we cannot prove by logic alone that 7+5=12. It is a mathematical fact that must be proved from mathematical axioms and we cannot "derive" it only by "inner intuition" about the idea of number. Commented Oct 18, 2018 at 12:49
  • okey, but still these axioms are laws of the system in which we do not discover (I mean get synthetic knowledge) anything new because the result is just from the law we set up. Commented Oct 18, 2018 at 12:58
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    7 + 5 = 12 is something that proceeds from a suitable set of axioms. Given other sets of axioms, it doesn't hold. For example, 7 + 5 = 2 mod 10. While it's true that , for many things, taking 7 of them and then 5 of them again yields 12 of them, that isn't always the case. If I remember my chemistry aright, 7 ml of water and 5 ml of alcohol doesn't yield 12 ml of fluid. There are things where 7 + 5 = 12 without question, but we have to find which things those are empirically. Commented Oct 19, 2018 at 17:30

1 Answer 1


"The analytic knowledge is known just by definition of the system we create" is more or less true under the modern conception of logic and deduction, largely developed due to the efforts of Frege, Peirce, Russell and others at the end of 19th, beginning of 20th century. It was not true on the conception of logic available in Kant's time, Aristotle's syllogistic being the paradigm. It only allowed a primitive form of conceptual inference, basically unpacking the definitions and applying simple syllogistic figures.

Kant was right to point out that theorems of Euclidean geometry do not follow from Euclid's axioms this primitive way. So he was right that mathematics is not analytic on his conception of "analytic", but not on Russell's much more expanded conception. Russell's writing is not particularly crisp on the difference, but he was aware of it as one of the architects of the shift. As he wrote in the introduction to Principles of Mathematics (1903, not to be confused with later Principia):

"It seemed plain that mathematics consists of deductions, and yet the orthodox accounts of deduction were largely or wholly inapplicable to existing mathematics... In this fact lay the strength of the Kantian view, which asserted that mathematical reasoning is not strictly formal, but always uses intuitions, i.e. the a priori knowledge of space and time."

Here is a more detailed explanation from Azzouni's Why Do Informal Proofs Conform to Formal Norms?:

"Analytic statements are to be recognized by an analysis of the concepts involved in those statements. It can be proven in Euclidean geometry that the interior angles of any triangle sum to 180◦. The proof proceeds, however, not by an analysis of a free-standing triangle (and its conceptually-given parts), but by embedding it within a larger figure. This is a ubiquitous and crucial aspect of mathematical proof: proofs are routinely facilitated (and often must be facilitated) by introducing concepts and methodology that — strictly speaking — go beyond (sometimes far beyond) the conceptual givens of what’s to be shown. What makes this a problem for Kant is that (for him and for his contemporaries) analyticity exhausts the resources of logic.

[...] Back then, the notion was restricted, pretty much, to an analysis of concepts almost along the model of Lego blocks. A concept — corresponding to a word — was taken to be composed of parts, and recognizable as so made up. This was why Kant’s examples of a priori synthetic truths, such as “7+5=12,” were successful counterexamples. It was presumed recognizable that the concept of “+”, for example, isn’t part of the notion of “12”. The contemporary notion... has additional resources."

However, Kant's notion of synthetic a priori is still non-empirical, unlike "synthetic" as it is largely taken to be today. This is because "a priori" is associated with conceptual/intellectual capacities only, if they are admitted at all. Kant, however, believed that we have an additional a priori faculty, the so-called "pure intuition", a kind of internal form of sensibility where the a priori synthesis of mathematics takes place. This is how he filled in the missing piece that is now fulfilled by far more expansive analytic use of definitions with the resources of propositional and quantificational logic. His solution was ingenious for his time, but it did not generalize well beyond arithmetic and geometry, the way modern one does. 4D space was a problem for Kant's original conception, just as non-Euclidean geometry was, although his successors came up with ways of extending "synthetic" to accomodate it. Azzouni again:

"It’s remarkable that this problem was first discovered by Kant, who wasn’t a professional mathematician (although, of course, he was familiar with mathematics)... It’s more remarkable that this “explanatory gap” wasn’t seen by Leibniz, a first-rate mathematician as well as philosopher, who — otherwise — thought deeply and far-sightedly about the nature of mathematical proof.

[...] The problem may be posed this way: a medium for proof is required that bridges the gap between the concepts involved in the statement of a theorem, and those that arise in a proof of that theorem. Kant, notoriously, hoped that intuition (of space and time) would fill that gap. This solution loses plausibility once the scope of mathematical proof is recognized to extend well beyond the traditional domains of geometry and number."

As for randomness, it is not relevant to the modern, empirical, notion of synthetic. That part of knowledge that necessarily involves external senses would be synthetic, whether the world is deterministic or not makes no difference. It is possible, of course, that God, or Laplace's demon, may be able to find out what goes on in advance by using deterministic laws, but they would have to know, analytically, that those laws hold. We do not.

Kant actually believed that we do know some of them a priori (Newton's laws), but that is a different story, and he believed that we know them synthetically a priori. Even when he speculates about God in Critique of Judgement he tends to suggest that God's intellect is intuitive ("archetypal"), and hence his perfect knowledge is all synthetic a priori. It is the wretched creatures like us who in addition need the analytic crutch of discursive intellect.

  • I didn't get "Kant was right to point out that theorems of Euclidean geometry do not follow from Euclid's axioms " but I have to spend more time to think about that. Thanks a lot for the answer! Commented Oct 19, 2018 at 10:54
  • @VostanMinor Last I checked Euclid, most of the proofs looked valid to me, in that they were valid deductions from the definitions, axioms, and postulates. There are a few problems, but mathematicians make mistakes in proofs all the time. I have no idea what significance it has that geometrical diagrams in proofs show things other than what's being proven. Commented Oct 19, 2018 at 17:24
  • @VostanMinor It is common after Hilbert when reading Euclid to substitute modern logical derivations for what he was actually doing. A now classical comprehensive study of Euclidean demonstrations is Euclidean Diagram by Manders, freely available in Mancosu volume:"With obvious exceptions, almost every step in traditional geometrical argument finds its license partly in the arrangement of the diagram".
    – Conifold
    Commented Oct 22, 2018 at 20:09
  • @DavidThornley Same, and there are almost no "mistakes" in Euclid. What is seen as "gaps" today is due to a different approach that accepts synthetic inference in addition to logical deduction. Euclid's demonstrations are not formally valid, nor are they meant to be.
    – Conifold
    Commented Oct 22, 2018 at 20:13

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