FOM: certainty; reply to Noria, Vorobey, Trybulec

Anatoly Vorobey mellon at
Thu Jan 7 09:02:08 EST 1999

You, Vladimir Sazonov, were spotted writing this on Tue, Jan 05, 1999 at 02:49:33AM +0300:

> > > Yes, this is a very good syntactic analysis of this big problem. What
> > > about semantics? Take, e.g. two drops of water (or vodka or what you
> > > like) + again two drops. The result will be 2 + 2 = 1 (one big drop).
> >

> > The point, to put it in plain words, is that there's nothing strange about
> > 4 raindrops merging into one, and it isn't at all relevant to natural numbers.
> > This particular "paradox" is mentioned, alas, all too often, probably due to its
> > cute and laconic way of arriving to a "contradiction". 

> It seems you took too seriously my joke. 

I didn't assume that your example was seriously meant; I simply used
the opportunity to clarify this example which is used seriously all too 
often and is always simply a distracting red herring. For one example
among a multitude of possible ones, here is Hersh writing in 
"What Is Mathematics, Really?", trying to explain different ways of 
justifying 2+2=4 and why they're all inadequate and only his "humanist"
philosophy of mathematics is up to the task:

  The most elementary answer is the empiricist one ."2+2=4" means "Put
  two buttons in a jar, put in two more, and you have four buttons in
  a jar." [...] Who knows if some exotic chemical reaction might give

      two buttons + two buttons = zero buttons


      two buttons + two buttons = five buttons.

  For indubitability, forget buttons.     

I.e. Hersh makes the mistake of confusing physical operation of
placing the objects spatially together with the mental one of
conceptualizing them as parts of one whole.

> Of course, pebbles are more appropriate than drops of a 
> liquid to fix the traditional meaning of addition operation for 
> natural numbers.  That example was used to show that we should 
> first fix some meaning and only then say "2 + 2 = 4 is true with 
> respect to this meaning".  

Not at all. If we were dealing with a formula 


then you could say that we first need to fix the meaning of f to
be the traditional addition operation, and the meaning of R to be
the traditional equality between sizes of sets, in order to talk
about its absolute truth. In case of the formula '2+2=4' these
meanings are assumed and that's the whole point of using + and =
instead of f and R. If the traditional + and = are well-defined for
2 and 4 and have absolute meaning, then what exactly is the problem
of saying "2+2=4 is absolutely true"? You could claim that the problem
lies with the traditional meaning, that it's not uniquely defined or
that it's vague, that different mathematicians mean different things
when they use + and = for concepts like 2 and 4 - and in that case 
"absolutely true" would indeed be wrong. But then *you* need to argue 
why it's ill-defined, and the "argument" of water-drops is simply
not adequate for the task, is logically irrelevant - that was my
original point.

Anatoly Vorobey,
mellon at
"Angels can fly because they take themselves lightly" - G.K.Chesterton

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