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David Godden's paper is primarily focused on two things. First, it attempts to
position Mill's work historically, coming sometime in the 19th
century placing it in and around Frege's work on logic and after Kant's
'Critique'. Second, the paper seeks to explain exactly how Mill's logic differs
and compares to those of its time, such as Whately's 'Elements of Logic'. The
most interesting points about Mill's work is what it says about his empiricism.
Mill criticized Whately on several points, including Whately's realist and
essentialist account of classification and definition, which conflicted with
Mill's own nominalism (9). What this points out to us is that Mill's own
philosophy on certain topics is a strongly empiricist one, most interestingly
relating to mathematics and logic.  Considering the arguments made by
rationalists from the 17th to the 19th century which made
mathematics out to be some sort of a priori enterprise, Kant's arguments at
least brought to question the widely accepted belief that mathematics was a
priori and analytic. Indeed, his arguments pointed out something that seemed to
be a synthetic nature of mathematical facts. Mill's work took this project a
step further, attempting to make mathematics entirely empirical and, indeed,
attempted to justify the basis for a synthetic mathematics. Considering my own
staunch opposition to the analytic/synthetic distinction I perhaps may be too
critical of Mill's project but I will try and temper my concerns. 

Mill holds that numbers have the remarkable peculiarity that they are
propositions concerning all things whatever; all objects, all existences of
every kind, known to our experience (22). Numbers are essentially about
'things'. Two, for instance, represents all pairs of things. For relatively
small numbers this seems like a clear point. Indeed, it seems to be an oft
brought up analogy for what numbers are. However, the analogy generally breaks
down when numbers become quite large. For instance, in videos about large
amounts of money (hundreds of millions of dollars, say), I have seen graphic
representations that demonstrate what this means in terms of something like
bananas being stretched around the Earth's equator. It aims to relate numbers to
something we have a reference for the general size of a banana
- and a rather large figure - the circumference of the Earth at its largest
  cross-section - to figures that are frequently taken to be beyond human
imagination.  The analogy attempts to clarify a fact, but still leaves something
to be desired. Indeed, Frege asks a very pointed question of Mill's
understanding of number, asking "what observed or physical fact is asserted in
the definition of the number 777864" (25). Frege also points out that "no one...
has ever seen or touched 0 pebbles" (25). Thus, it seems that Mill almost misses
the point about what a number is in attempting to reduce them to representations
of things.  Perhaps Mill would argue that what really matters about the science
of number is that it is sufficient to understand small numbers like two or
twelve through empirical means, and via a principle of induction we can extend
this understanding to other numbers like zero or 777,864. As induction isn't
wholly a logical principle on Mill's view, it's hard to understand how induction
may serve to ground this understanding or if it justifies the point on what a
number is, but instead represents some kind of ad hoc saving of synthetic

As an aside to all of this, I have a sort of general question about the
synthetic/analytic distinction. To what extent are these related to contingency
and necessity? It seems that analytic facts are those which are necessary and
synthetic those which are contingent. Is this all that the distinction amounts
to? Are synthetic propositions nothing over and above contingent facts?


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