FOM: Tennant's re-invention of the numbers
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Mon Feb 2 14:29:18 EST 1998
Sol Feferman writes:
> (American Heritage Dictionary, 3d edn.):
> "Invent, tr.v. 1. To produce or contrive (something previously unknown) by
> the use of ingenuity or imagination. 2. To make up, fabricate."
>
> Neil, what definition do you use? Perhaps Wigner didn't mean what he said,
> but taken at face value it's hard to read it other than the way that I
> did.
[Wigner had said:
> mathematics is the science of skillful operations
>with concepts and rules invented just for this purpose. The principal
>emphasis is on the invention of concepts.
(NB: no mention by Wigner of subjectivity)
and Feferman had commented:
>This seems to point to the subjective origin of mathematical concepts
>and rules.
To this I had replied:
>Why should the invention of concepts be a *subjective* matter?]
Contrary to what Sol assumes about the dictionary definition, I contend
that it does *not* legitimate any inference to the *subjectivity* of
either what is invented or the process of inventing it. Concepts and rules
are public and communicable: at least inter-subjective, and perhaps even
objective in a yet stronger sense than "inter-subjective".
I can produce or contrive something previously unknown by the use of
ingenuity or imagination---say, a new axiom A that settles CH---and I can
make up or fabricate a proof of CH from ZFC+A, without anything that I
do (make up, imagine) deserving to be called *subjective*. It's just
*intellection* that is going on here, not anything in the way of
subjective experience.
With true subjectivity of *experience*, we have first-person privileged
access (only the experiencing subject really knows the precise content
of his or her subjective experience) and varying degrees of ineffability
(certain subjective experiences are so fine-tuned and nuanced that it is
impossible to capture them precisely in words, or to convey their precise
content to others). With true subjectivity *of opinion* (directed to a
communicable content) we have the idea that there is no definitive fact
of the matter for the opinion to reflect, or be held answerable to;
rather, the "fact" of the matter is (if it exists at all) somehow
constructed out of best opinion on the matter.
None of this characterizes mathematical concepts *even when they are
invented*. For (pace Brouwer) they are invented *for the community*,
and are *communicable*; and true propositions constructed out of them
answer to facts of the matter, i.e., to proofs by communally accepted
methods from communally accepted axioms.
Now I grant Sol his observation that something can have a subjective origin
without necessarily being subjective. He wrote:
> Note that I spoke about the subjective *origin* of mathematical
> concepts. This does not imply that once invented, mathematical concepts
> have no objective status.
But my query did not depend on any failure to appreciate that point.
My query concerned the alleged *subjective* origin of mathematical concepts.
I wanted to point out that *even in their origins*, mathematical concepts
are objective (if anything is). For, mathematical concepts arise from new
parsings of thoughts. The thoughts are objective; their new parsings are
objective; and thus the concepts arising from those parsings are objective.
(Examples of parsings: "There are exactly two apples" becomes "the number of
apples = 2"; "line segment A is twice as long as line segment B" becomes
"the length of A = 2 x the length of B"; "a is F, b is F, c is F, and
nothing else is F" becomes "the set of Fs = {a,b,c}"; etc.)
Feferman went on to write (and a quote of this length is useful, since
it says in places what I would want to say anyway):
> The main part of Tennant's riposte is devoted to a neo-Fregean
> re-invention of the number concept. Basically this comes to the
> (contextual) introduction of such numbers as 2 in a conservative extension
> of logic, with #xFx = 2 equivalent to "there are exactly two F's". Is
> this really a re-invention? More importantly, is this an explanation of
> the number concept? NO. It is only a partial explanation for each
> natural number n, as to how the equation #xFx = n may be reasoned with.
> It does not explain the general notion of natural number. If that is to
> have the properties of Peano's axioms for 0 and successor, with induction
> expressed for arbitrary properties, much more has to be done. Doing so
> may indeed "command rational assent", but first one has to have the idea
> how that is to be done (Dedekind, Frege being two approaches), then it has to
> be communicated.
The account I sketched was not simply a "partial" (i.e. non-total) explanation
of the number concept. I referred the reader for further details to my book
Anti-Realism and Logic, where the Peano-Dedekind axioms are derived in full
rigor, using only intuitionistic relevant logic. The proposed rules for
"zero", "successor" and "the numbers of..." yield the Peano-Dedekind axioms
as part of their constructive content. (These axioms involve the general
notion "...is a number".) The proposed rules *also* yield the conceptual
control
there are exactly n Fs iff #xFx=n*
which was arguably Frege's *objective* intellectual origin of the whole
account, and a constraint which, in my view, is too often neglected in
foundational accounts of the natural numbers.
Neil Tennant
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