[FOM] Why Mathematics needs verbs

[Dean Buckner]

In a discussion group concerned with the foundations of mathematics, perhaps
I will be allowed to justify why verbs, which seem to be connected with the
science of grammar, are also important for mathematics.  Consider

    S said that 2 + 2 = 5

So long as S did in fact say this, the sentence is true.  But it contains
the string of symbols "2+2=5", which if uttered on its own, and if the
symbols had the same meaning they ordinarily have, would express something
false [1].

Very often we need to talk about what someone said, without saying it
ourselves, particularly if we are dubious about the truth of what they say.
Hence the celebrated "that" clause, which is simply a means of removing the
assertion bit of a sentence, without removing anything else.

But as soon as we allow "that" clauses, i.e. as soon as we allow the device
of talking about the meaning of a sentence, we require verbs.  Here's why.
Suppose instead of  using the word "that", we enclose a sentence inside
brackets as follows

    (2 + 2 = 4)

and so can say S thinks (2 + 2 = 4), or (2 + 2 = 4) was proved, and so on.
But suppose instead of talking about this meaning, I want to express this
meaning itself.  It's no good my saying that (2 + 2 = 4) is thought by me
(if I really thought that 2 + 2 = 5, I would be lying).  So let's invent the
word "true" which simply does the job of removing the brackets round the
sentence, and so equivalent to uttering "2 + 2 = 4".  so I utter

    true (2 + 2 = 4)

Now this has a completely different meaning from the bracketed sentence
without the word "true".  We agree?  But if it has a completely different
meaning, what is that meaning?  If our language allows us to talk about
meanings, we can't we give a meaning to what is created when we add "true"
to "(2 + 2 = 4)"?  Well. let's try wrapping the "true" expression itself in
brackets, thus

    (true (2 + 2 = 4))

But this won't work, for "true (2 + 2 = 4)" is equivalent to "2 + 2 = 4",
thus, substituting

        (true (2 + 2 = 4)) = (2 + 2 = 4)

whereas we wanted to get the meaning of the expression containing "true" on
the outside.  The fact is, the function of the brackets is to designate the
meaning of the words inside the brackets, the function of the word "true" is
to remove them.  Thus, the word "true" has a perfectly good semantic
function, a meaning if you like, but it is impossible to put a name to it
[2].  And that's why verbs exist.  They are a semantic contribution to a
sentence that cannot be given by any name, for concerning any name "N" we
can always write ""N" names N".  We cannot do the same with a verb.

Verbs also exist in mathematics, believe it or not.  The expression "2 + 2 =
4" has a verb built into it (inside the identity sign in fact).  And that's
why verbs are important for mathematics as well as grammar.


Dean



[1] Compare Philosophical Investigations ~22.  "Frege's assertion sign marks
the _beginning of the sentence_.  Thus its function is like that of the
full-stop.  It distinguishes the whole period from a clause within the
period.  If I hear someone say "it's raining" but do not know whether I have
heard the beginning and end of the period, so far this sentence does not
serve to tell me anything".  Cf Zettel ~684.

[2] Frege got tremendously puzzled by this.  "The word "true" seems to make
the impossible possible: it allows what corresponds to the assertoric force
to assume the form of a contribution to the thought" (My Basic Logical
Insights, Nachgelassene Schriften p. 272)  That is because Frege began the
tradition of regarding sentences as functions, i.e. as names of some kind.



Dean Buckner
London
ENGLAND

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