Logicism and Neo-Logicism: Handout 1
31 January 2006
What is logicism?
Overview
The problem. Pure mathematics appears to be very different from empirical science:
- Epistemologically, it appears not to rely on observation and experiment;
- Metaphysically, it appears to discover necessary truths that are concerned with
abstract objects.
The proposal. Logicism is the proposal that pure mathematics is reducible to logic:
- The basic concepts of mathematics are definable in purely logical terms.
- The truths of mathematics (expressed using these definitions) are provable by
pure logic alone.
Week 1. What is logicism? What might its philosophical payoff be?
Week 2. Kant’s philosophy of mathematics; the “arithmetization” of analysis
Week 3. What is “pure logic”? The status of second-order logic
Week 4. Ascriptions of number
Week 5. Numbers as objects
Week 6. Russell’s paradox
Week 7. Frege’s Theorem
Remaining weeks: critical evaluation of neo-logicism. Topics include
- Do we think of the natural numbers primarily as ordinals or as cardinals?
- Implicit definitions: Is there any default entitlement to lay down such definitions?
- The problem of logical objects.
- Ontological categories.
- Beyond arithmetic
Kant on the distinction a priori/a posteriori
A priori = independent of experience; a posteriori = dependent on experience.
But what sort of dependence?
- Not “in the order of time,” since “all our knowledge begins with experience.”
- Rather the sources of the concepts or knowledge: our cognitive faculty or
experience.
Characteristics of a priori knowledge
- If we know that p is necessary, then we know this a priori.
- If we know that p is “strictly universal,” then we know this a priori.
These characteristics reveal a variety of examples of a priori concepts and
knowledge: all of mathematics, the principle of causality, the concept of substance.
Kant on the distinction analytic/synthetic
A judgment is analytic iff the predicate concept is contained in the subject concept.
A judgment is synthetic iff this isn’t so but rather the predicate concept adds
something to the subject concept.
Examples: “Bodies are extended” is analytic; “Bodies are heavy” is synthetic.
Four possibilities
Analytic a posteriori: no such knowledge.
Analytic a priori: conceptual knowledge (cf. Hume’s “relations of ideas.”)
Synthetic a posteriori: empirical knowledge (cf. Hume’s “matters of fact.”)
Synthetic a priori: yes (the distinctive feature of Kant’s philosophy)
1.
Arithmetic. To find out that 7+5 = 12 we have to go outside the concepts and
call in the aid of the intuition which corresponds to one of them, e.g. fingers or
points. So arithmetical truths props are synthetic.
2.
Geometry. It is synthetic that the shortest line between two points is straight.
For my concept of straight contains nothing about quantity, only quality.
Intuition must thus be called in.
Frege’s claims about Kant
1. Kant is right that geometry is synthetic a priori.
2. Kant was wrong about arithmetic, which is really analytic.
3. By analytic/synthetic and a priori/a posteriori, Frege means the same as Kant.
Frege’s definitions of the epistemic categories
A truth is a priori iff “its proof can be derived exclusively from general laws, which
themselves neither need nor admit of proof” (FA §3); otherwise the truth is a
posteriori.
A truth is analytic iff, “in carrying out [the] process [of finding the proof of the truth
and of following it up right back to the primitive truths], we come only on general
logical laws and on definitions” (FA §3); otherwise it is synthetic.
(Quine’s refinement of Frege’s definition: a truth is analytic iff it can be turned into a
truth of logic by substitution of synonym for synonym.)
Differences between Frege’s definitions and Kant’s
a.
Frege de-psychologizes much of what Kant talk about: He doesn’t talk about
subjective ideas but of objective contents of judgments.
b.
Instead of talking about the sources of knowledge, Frege talks about the best
possible proof of propositions.
c.
Frege’s logic is much richer than Kant’s outdated Aristotelian logic. This leads
to a much larger class of analytic truths.
Why arithmetical truths may be analytic after all
1.
The “arithmetization” of analysis gives reason to be optimistic. (FA §1)
2.
Analytic truths holds of everything that can be thought at all, whereas
synthetic a priori truths hold only of what is intuitable. The truths of
arithmetic appear to have the same range of validity as analytic truths. (FA
§14)
But to conclusively establish that arithmetic is analytic, we need proofs!