, 2005, 2006), and Lisa Sha-
bel (1998) have done important work on the historical context that informed
Kant's understanding of mathematics. They have shown that in addition to
Euclid's writings on geometry and the general theory of arithmetical and geomet-
rical proportions, Kant's understanding of
example of Euclid’s axiomatic treatment of geometry, the axioms of a theory are
truths about the subject matter of the theory which are so basic that they entail
everything the theory has to say about its subject matter (and thus are ideally suited
to organize the body of knowledge constituted by
suitable variable, or if there are too few. Take for example Euclid's statement
 There are infinitely many prime numbers
and assurne it is true. On the face-value interpretation this requires the exis-
tence of infinitely many prime numbers. On the substitutional interpretation,
for it to have the
But there are pertinent examples that suggest otherwise (see Baker/Hacker 1985:
ch. VIII). Statements that were once regarded as paradigms of necessary truth,
such as Euclid’s 5th axiom or the claim that there cannot be negative numbers have
not just been stripped of their necessary
einer Definition erforderlich ist. Euclid’s
Definition von Parallellinien ist von der Art.
1Anmerkung in der Akademie-Ausgabe. 2Bei Textpassagen mit vorangestelltem 𝑔 handelt es sich um Zusätze
Kants aus derselben Schriftphase. 3In eckigen Klammern [ ] stehen von Kant ausgestrichene Wörter und
same figure, a circle, but whereas the first is simply a genetic version of Euclid’s definition and his corresponding
postulate, Euclid demanded a demonstration in the case of the second. (See On Universal Synthesis and Analysis,
or the Art of Discovery and Judgment [PPL 230].)
40 P 4:316, KrV
Whittaker, E. (1958) From Euclid to Eddington, New York: Dover Publications.
Wilson,]. (2006) Jorge Luis Borges, London: Reaktion Books.
Wingenstein, L. (1993) Tractatus Logico-Philosophicus, London: Routledge.
Wingenstien, L. (1958) Philosophical Investigations, Oxford: Basil Blackwell.
Zammito, ]. H
proved in this way.
The second way is to show “how such a thing could come into being”, since “if we know
how something could come into being, we may no more doubt as to whether it could be”.
Wolff explains that Euclid “proves in this way that an equilateral triangle is possible when
he shows how [such a
AnPo 1.4-5, see 1.4, 73b32-4 and my 2006a.
322 Pieter Sjoerd Hasper
Acerbi, F. 2008. Euclid's Pseudaria. Archive o[ the History o[ the Exad Sciences 62, 511-551.
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