According to Locke, it is impossible to obtain substantive knowledge from analytic propositions. Statements like "triangle has three sides" are analytic, but one cannot derive the Pythagorean Theorem analytically. However, Frege says that mathematical (arithmetical) truths are analytic. This raises a problem that although Locke's thesis about analytic knowledge is very convincing, analytic arithmetic knowledge is possible. How is this possible?

  • Kant has handled this issue, and argues that mathematical knowledge is synthetic instead of analytic, aiming. Just some food for thought
    – Cicero
    Commented Jun 10, 2015 at 22:52
  • but doesnt his notion of space and time wrong?
    – Tom
    Commented Jun 10, 2015 at 23:06
  • 1
    his views of space and time are irrelevant. And who established that his notion is wrong (people may not agree with them, but they certainy have not been refuted completely).
    – Cicero
    Commented Jun 10, 2015 at 23:08
  • Hello. What is the source of what you say about Locke? Commented Jun 10, 2015 at 23:18
  • 1
    It would be nice to have a source for this; as generally I associate this distinction with Kant rather than Locke; if you browse the site you will find some refutations of the simple critiques of Kantian space-time Commented Jun 12, 2015 at 17:00

2 Answers 2


Frege is the founder of a program called logicism that aimed to reduce all of mathematics to logic. In order to reduce mathematics to logic Frege had to expand what is meant by logic. Before him Locke, Kant and others understood by logic only Aristotle's syllogistic, which is a manipulation of simple implications (syllogisms). Frege's Logic went much further, and encompassed all of arithmetic in particular, if not all of set theory. But derivable from logic alone is what all three meant by "analytic". Under logicism all of mathematics would be analytic, including much of what Kant called "synthetic a priori" (but not the parts related to geometry and physics). So both Locke and Frege were right, they just meant different things by "analytic".

"Frege's conception of the analytic was suitably broader than Kant's. Kant required that conceptual containments be evident within the sentence, rather than that the sentence be displayed as a conclusion following logically from axioms whose own logical or conceptual truth was self-evident, and which might contain expressions not occurring in the sentence in question... Kant did not regard ‘7 + 5 = 12’ as an analytic truth. The Fregean, by contrast, is able to exploit the internal structure of the numerals, and to invoke the recursion axioms for addition (which themselves would have to have been derived in logicist fashion). So, for the Fregean, even if not for Kant, ‘7 + 5 = 12’ is an analytic truth."

The best attempt to realize the logicism program was Russell and Whitehead's book Principia Mathematica. However, after Gödel's results on incompleteness the program came to be viewed as a dead end, and was largely abandoned. Later, Quine argued convincingly that the analytic/synthetic distinction itself, even in a revised form adopted by logical positivists, can not be maintained at all. According to Quine, all knowledge, including the laws of logic, is synthetic, and ultimately empirical, this dealt a further blow to logicism. Although it is no longer believed that "all of mathematics" reduces to logic, the modern mathematical logic is much closer to Frege's conception of logic than to Kant's or Locke's.


Frege's thesis was not that mathematics as a whole was analytic, just that arithmetic (the theory of whole numbers) was so. Frege criticized Kant about arithmetic, but he agreed with Kant that geometry was synthetic. As to that Pythagoras' theorem cannot be deduced from the definition of a triangle alone, there is surely no dispute.

We shall do well in general not to overestimate the extent to which arithmetic is akin to geometry . . . For purposes of conceptual thought we can always assume the contrary of some one or other of the geometrical axioms, without involving ourselves in any self-contradictions . . . The fact that this is possible shows that the axioms of geometry are independent of one another and of the primitive laws of logic, and consequently are synthetic. (Frege, The Foundations of Arithmetic §13)

  • I agree with the point, but I want to point out that Frege's reasoning here is questionable. "For purposes of conceptual thought" we can, and do, assume the contrary of the axioms of arithmetic as well, without self-contradictions. Kant presented arithmetic as a priori synthesis in time, and geometry as a priori synthesis in space. He may have been wrong about space and time as forms of a priori perception, but he was not wrong about the affinity of arithmetic and geometric reasoning. Quine showed that both Kant's and Frege's lines between analytic and synthetic are drawn by convention alone.
    – Conifold
    Commented Jun 13, 2015 at 0:25
  • @Conifold Hi. 1) Where do we "assume the contrary of the axioms of arithmetic as well"? 2) How do you know that Kant "was not wrong about the affinity of arithmetic and geometric reasoning"? The relation between geometry and space is obvious. The relation between arithmetic and time is not obvious, and many (like Frege) did not acknowledge it. Commented Jun 13, 2015 at 17:48
  • It turns on "assume" and "obvious". We can reason about non-Euclidean geometry, we can visualize Euclidean models of it, even form mental intuition on how it works, but we can do all that with modular arithmetic too. The process Kant describes of invoking schemes of 7 and 5 and counting 7+5 as 12 is akin to invoking the scheme of triangle and mentally manipulating it. Either we schematize the experience of counting empirical objects or 7+5 can be 2 mod 10, and counting is related to time as obviously as shapes to space.
    – Conifold
    Commented Jun 13, 2015 at 20:26
  • But even logic proper is not immune to assuming otherwise, intuitionists rejected excluded middle, Heraclitus even rejected a=a with "you can't enter the same river twice". And Frege's Logic reached well beyond arithmetic of whole numbers, indeed he ran into trouble with Russell's paradox because unrestricted comprehension for classes was "obvious" plato.stanford.edu/entries/russell-paradox/#HOTP Quine's point was that "obvious" is in the eye of the beholder and of impure a posteriori origins, nothing is truly analytic.
    – Conifold
    Commented Jun 13, 2015 at 22:08

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