[FOM] A Defence of Set Theory as Foundations
Robert Lindauer
rlindauer at gmail.com
Sat Oct 15 03:21:55 EDT 2005
On Oct 13, 2005, at 2:56 PM, Nik Weaver wrote:
> Robert Lindauer wrote:
>
>> The notion of a "constructed" infinite group is, I think, incoherent.
>> The operational nature of constructing things leaves us a limited
>> amount of time in which to do it, leaving us a limited number of
>> things
>> constructed.
>
> I suppose so, if you take "constructible" to mean "physically
> constructible in our universe". No doubt the question of what
> is actually physically possible is of the highest scientific
> interest, but I don't see it as particularly relevant to the
> fundamental nature of mathematics. I would say that in math
> we study the realm of logical possibility, not physical
> possibility.
>
> Or do you mean that a construction of length omega is not
> even logically possible? (You use the word "incoherent".)
> My position is that we can form a definite picture of an
> imaginary world in which constructions of length omega can
> be carried out, and this is sufficient to justify their
> status as logical possibilities.
>
> I think most mathematicians would agree with that --- it
> is on this kind of basis that I believe most mathematicians
> accept that all arithmetical statements have a definite
> truth-value. We believe the twin primes conjecture is
> definitely true or false because we can imagine running
> through the natural numbers and checking it.
>
> This is the conceptualist view.
I think one has to have a clearer sense of foundations of modal logic
to really tackle this in detail, but let's parse it against what I
wrote - e.g. the "inaccessible platonic world" of which we have
knowledge by some "direct intuition" or perhaps some spiritual
connection. Until there is a good way of communicating how some people
seem to have this intuition of other possible worlds where other people
don't have the same intuitions, the whole prospect has to look too
dubious for mathematics, eg:
1 + 1 = 2 because we can imagine a possible world such that....
seems much less accessible than simply:
1 + 1 = 2 because, for the most part, ordinary medium-sized physical
objects in our world when combined into groups tend not to morph into a
single thing and giving us "1 + 1 = 1" nor do they tend to multiply
inexplicably giving us "1 + 1 = 3", etc.
In fact, the possible worlds are less accessible than simply believing
in the Platonic World, since both have the same inaccessible nature -
how do our physical minds get their grips on these extra-worldly
objects? If our minds are spiritual, how do our spiritual minds get
their grips on these extra-worldly objects and why is it that some
people seem to grasp them very differently and with the same vehemence?
Somehow, though, to me the platonist sounds more reasonable, I'm sure
others have other feelings about it.
Personally, I can't imagine the world where someone actually checks the
twin prime conjecture by running through all of the integers and
checking whether each was divisible, etc. I can say the sentence, but
wouldn't know how to recognize something as "having run through all the
integers". You apparently have a more vivid imagination than I do.
This is to say, while I have some kind of idea of what running through
the integers looking for primes is like, I have no idea what running
through ALL of them would be like, and certainly not doing all the
division operations required. For instance, I can imagine the size of
a machine that could divide integers with 10,000,000 digits with one
cycle in them by doing some arithmetic and having a basic idea of how
small we can make a transistor, etc. But I can't begin to imagine what
it'd be like to divide an integer with arbitrarily many digits, and
certainly not what the underlying physical structure of such a thing
would have to be. Perhaps only a god could do such a thing (assuming
that the idea of doing such a thing makes sense in the first place,
which is what I doubt.) Can I imagine olympus? Should our mathematics
turn on that kind of question?
I suggest what you may be imagining is running through the integers,
but surely you're not imagining actually finishing the process, right?
In any world in which things must be done "one at a time" such a thing
is ludicrous. I'm not sure what the worlds in which things are done
transfinitely are like, nor do I have any idea what they might be like
- worlds in which one apprehends an infinite landscape in a single
gestalt, etc. But I genuinely can't imagine lifting a rock who's
weight was transfinitely many pounds, or crossing a bridge that was
transfinitely many yards across.
I simply don't know what to imagine when those kinds of things are
said. This is because I can't really imagine the process that would
achieve such things, perhaps you could offer something more descriptive
to help me imagine such a process?
You might say "I have a different meaning for "construct" in mind."
Very well. It's the sense in which I could instantaneously create an
infinite universe and in the next instant destroy it. But I don't know
what it's like to be God, really, and even He took 7 days, as the story
goes.
> I have a hard time with this. Does the story have to be
> consistent?
I think the right question is "consistent according to who?" or
"consistent with what?"
One can tell a story that's "internally consistent" by simply defining
the term "consistency" within the story so that it has the properties
required for what the story calls "consistent". This doesn't make the
story -really possible-, and maybe not even "really consistent".
An example may be in order:
"Robbie was talking about a kind of mathematics in which there exists
some number x such that x + 1 = x, and x - x = 0, and 0 = 1, and it was
consistent."
Someone else might regard such a story as incoherent straight off
because they might think that the possibility of some x + 1 being equal
to x is incoherent (like me, for instance). But they wouldn't be
required to find a specific contradiction in the story because, after
all, consistency is defined in the story as allowing 0 = 1. When the
person complains "that's a contradiction" the teller says "not
according to this definition of inconsistency". The saying goes that
the history of math is the story of one-time contradictions becoming
axiomatic. Perhaps as our lifetime and leisure time increases so does
our communal capacity to imagine.
"0 = 1 in his story" would be our report, and we'd just note that they
had a different sense of consistency than we did. Maybe we'd get a
good chuckle.
Lots of incoherent stories are internally coherent - IF you define
what coherence is in the story. In normal fictional stories this is
called "suspended disbelief". When a schizophrenic tells a story, the
psychologist, in order to identify with them, has to suspend their
disbelief in order to understand the experience that the patient is
describing. It becomes trickier when one compares formal systems.
One CAN "justify" anything this way - if one regards internal
consistency as a sign of import. I would just suggest that the use of
the term "justify" here is much, much too strong. Perhaps "derive" or
"imagine" would be better.
Finally, I don't see why one has to imagine another world in order to
recognize that 1 = 1. 1= 1 in our world, doesn't it? Those are the
kinds of intuitions upon which actual mathematical practice are based:
1 = 1, 1 + 1 = 2, 3 follows 2 when you count, etc.
Robbie Lindauer
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