So my first draft addressed many of the points you made, but it that
email got too big and sprawling I thought.
So I've focused on what seems to me like the key passage from your
post. If you think there was some other point that I should have
addressed, let me know.
So, key passage:
> Do these mathematical objects "really" exist? I'd say they have
> logico-mathematical existence, not the same existence as tables and
> chairs, or quarks and electrons.
So which kind of existence do you believe is more fundamental? Which
is primary? Logico-mathematical existence, or quark existence? Or
are they separate but equal kinds of existence?
In what way, exactly, does logico-mathematical existence differ from
quark existence? Is logico-mathematical existence a lesser kind of
existence? Is logico-mathematical existence derivative of and
dependant on quark existence?
Further, do tables and chairs even have the same kind of existence as
quarks and electrons? A table is something that we perceive visually,
but we intellectually take "tables" to be ultimately and fully
reducible to "quarks and electrons". So chairs and quarks certainly
exist at different levels. Quarks would seem to be more fundamental
than chairs. But obviously we don't actually perceive quarks or
electrons...instead we infer their existence from our actual
perceptions of various types of experimental equipment and from there
associate them back with tables.
As for our experience of logico-mathematical objects, we certainly can
translate them into more "chair-like" perceptions by visualization via
computer programs, right? This would put them very much on similar
footing with our experience of quarks and electrons at least, which we
also only visualize via computer reconstructions.
And, presumably it is possible for a human with exceptional
visualization abilities to experience logico mathematical objects in a
way that is even more "chair-like" than that. For instance, there are
people with Synesthesia (http://en.wikipedia.org/wiki/Synesthesia),
for whom some letters or numbers are perceived as inherently colored,
or for whom numbers, months of the year, and/or days of the week
elicit precise locations in space (for example, 1980 may be "farther
away" than 1990).
But what if this type of synesthesia had some use that strongly aided
in human survival and reproduction? Then (speaking in materialist
terms) as we evolved synesthesia would have become a standard feature
for humans and would now be considered just part of our normal sensory
apparatus. We would be able to "sense" numbers in a way similar to
how we sense chairs. In this case we would almost certainly consider
numbers to be unquestionably objectively real and existing. Though
maybe we would ponder their peculiar qualities, in the same way we now
puzzle over the strangeness of quantum mechanics.
A further example:
"Autistic savant Daniel Tammet shot to fame when he set a European
record for the number of digits of pi he recited from memory (22,514).
For afters, he learned Icelandic in a week. But unlike many savants,
he's able to tell us how he does it.
Q. But how do you visualise a number? In the same way that I
visualise a giraffe?
A. Every number has a texture. If it is a "lumpy" number, then
immediately my mind will relate it to other numbers which are lumpy -
the lumpiness will tell me there is a relationship, there is a common
divisor, or a pattern between the digits.
Q. Can you give an example of a "lumpy" number?
A. For me, the ideal lumpy number is 37. It's like porridge. So 111,
a very pretty number, which is 3 times 37, is lumpy but it is also
round. It takes on the properties of both 37 and 3, which is round.
It's an intuitive and visual way of doing maths and thinking about
numbers. For me, the ideal lumpy number is 37. It's like porridge."
I think we can say (again, speaking in materialist/physicalist terms)
that it's purely an accident of evolution that numbers don't seem as
intuitively real to us as chairs, or colors, or love, or free will
Speaking in platonist terms, it's an accident of our particular
mental/symbolic structure that numbers don't seem as intuitively real
to us as chairs, or colors, or love, or free will (ha!).
Speaking in computationalist terms, it's an accident of our
causal/representational/algorithmic structure that numbers don't seem
as intuitively real to us as chairs, or colors, or love, or free will
But, no matter what terms you use, it's conceivable, and we have
significant evidence that points to the possibility, that our
conscious perceptions could be modified in a way such that numbers and
other abstractions would seem much more substantial and real than they
do currently, even as substantial and real as chairs and tables. And
this wouldn't require any change in what actually exists or "how"
these things exists (logico-mathematical or otherwise).
So based on all of the above, returning to your original statement:
"I'd say they have logico-mathematical existence, not the same
existence as tables and chairs, or quarks and electrons."
I would say that most people PERCEIVE logico-mathematical objects
differently than they perceive tables and chairs, or quarks and
electrons. But this doesn't tell us anything about whether these
things really have different kinds of existence. That we perceive
them differently is just an accident of fate.
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