[FOM] Natural Language and Mathematics (reply to Harvey)

[Harvey Friedman]

>My background is the philosophy of language.  This requires a training that
>encourages existential conservatism: to prefer for example to explain "there
>are no unicorns" as the negation of "there is at least one unicorn" i.e. as
>denying the existence of something, rather than asserting of any existing
>thing
>(the set of unicorns e.g.) that it is empty.  Frustra fit per plura quod
>potest fieri per pauciera (Ockham).

In any standard treatment of f.o.m., one also prefers to formalize 
"there are no whatevers" as "for all x, x is not a whatever", or "not 
(there exists a whatever)". Here there is no difference. Of course, 
the apparatus for doing it via the empty set exists in mathematics.
>
>So one "substantive issue" is set theory itself, which postulates the
>existence of things (sets) which seem unnecessary to explain the logic of
>our ordinary numerical statements such as "if there exists one thing and
>another thing, there exist two things, and if there is a third thing, there
>are three things".  I don't see why we need entities like sets to explain
>these sorts of statements.

We don't. However, the interesting question is: can you develop even 
elementary mathematics without significant existential commitments?

Any talk of existential commitments being incoherent or obviously 
wrongheaded is, at least on the face of it, silly.

>
>It has been argued here that formal discourse is a wholly different from
>ordinary discourse.  But the philosophy of language is not concerned with
>symbols or utterances, it is concerned with their meaning.  And I don't see
>that what ordinary people mean by their numerical discourse is any different
>from what mathematicians mean.  And if it is wholly different, what are
>mathematicians talking about?


Ordinary people in ordinary activity are not doing science, which 
requires a rather extraordinary degree of concentration and attention 
to detail. Again, any suggestion that mathematicians have no idea 
what they are talking about is, at least on the face of it, silly.

>
>Another substantive issue is the difference between the semantics of
>ordinary language, and mathematical sentences.  Ordinary language seems
>on the whole to prefer a semantics "all on one level".  For example
>
>(a) ordinary sentences are able to assert the truth of what they express (or
>rather, the expressing of what they express is one and the same with the
>assertion of truth).  So there is no need for the infinite Tarsksian
>hierarchy required to explain the assertion of "grass is green".  Nor can
>the Goedel phenomena appear in the sematnics of ordinary language.

It would be interesting to see if mathematics could be founded on an 
"all on one level" conception. I am still a bit vague on just what 
would count as a solution.

For example, according to conventional wisdom, mathematics needs, in 
an essential way, a system of objects called natural numbers, with a 
successor relation. Is this "all on one level"? Note that the 
successor relation can be criticized as not being on the same level 
as the natural numbers themselves.

The Godel phenomena of course appear in the semantics of ordinary 
language if that semantics interprets very weak fragments of 
arithmetic - which it arguably does.
>
>(b) there is a rigid distinction in ordinary language that forbids any kind
>of "comprehension" i.e. that forbids us reading "Socrates is bald" as a
>relation between Socrates and some entity (the entity <is bald>).  This
>makes the whole concept of a limit ordinal impossible.  So, no set
>hierarchy: it's finite ordinals "all the way up" (or rather, all the way
>across).

In other words, natural language forbids consideration of relations 
between objects as themselves objects?

How do you want to handle natural numbers, rational numbers, real 
numbers, all together? Also functions from any one of these three 
into any one of these three?

>
>Martin Davis (and others) have argued that this is because of a deficiency
>in natural language.

Deficiency for the foundations of mathematics is what is at issue.

>This view reflects the great schism between philosophy
>and logic that happened in the late nineteenth century, between the ideas of
>those who wanted to explain the way we ordinarily think, and who turned to
>natural language for this, and those like Frege who believed that natural
>language was inherently imprecise, and that true logic was only to be found
>ideal or artificial languages.

There is foundations of mathematics, foundations of science, on the 
one hand, and foundations of informal discourse on the other. 
Obviously they have quite different aims. The question on the table 
is what they have to do with each other, if anything at all.

>
>It's the existence of this divide which makes it harder than anything else
>to have a meaningful debate in F.O.M. which is largely dominated by the
>believers in ideal language.

Debate about what? I haven't seen any debate. There might be one if 
you were to assert any of the following:

1. That mathematics cannot be done under constraints you cite from 
ordinary informal discourse.

2. That mathematics can be done under constraints you cite from 
ordinary informal discourse.

3. That mathematics should be done under constraints you cite from 
ordinary informal discourse.

4. That mathematics must be done under constraints you cite from 
ordinary informal discourse.

Just reiterating, rather informally, what some of these constraints 
are, is not very productive. What would be productive is a 
sufficiently clear idea of what they are so that one could decide the 
truth values of 1 and 2.