Rex Allen wrote:
> 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?
The way I look at it there is knowledge we gain from perception, including the
inner perception of logical and mathematical facts. We make up theories that
unify and explain these perceptions and which extend beyond what we perceive.
These theories have ontologies: things they assume to exist - within the domain
of the theory. There's no way to say that one is more fundamental than the
other so long as they are in separate theories. Only if they are subsumed
within one theory can there be some sense in which one is more fundamental than
the other. I don't think we have such a theory yet. And note that even if we
have theory including both mathematical and physical objects in its ontology it
may turn out that either one can be used to explain the other; so it's not
necessarily the case that one is more fundmental.
> In what way, exactly, does logico-mathematical existence differ from
> quark existence?
You can kick quarks and they kick back.
>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?
Although the explanation of the macroscopic world from the quantum world is not
worked out it is generally supposed that tables and chairs will eventually be
explained in terms of quarks and electrons. The interesting thing is that from
the standpoint of epistemology, the tables and chairs are more fundamental,
while the theory makes the quarks and electrons more fundamental to the
ontology. So there are different senses of "fundamental" too.
>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?
I'm doubtful of that. Certainly many mathematical objects can be illustrated
because they were invented to describe something we could perceive - like
spheres or symmetries. But I don't see how you would visualize Shannon
information or strings in ten dimensional space.
>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.
But there's more than visualization. We can also manipulate and use quarks and
electrons, i.e. we can make them kick each other and us.
> 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).
I don't think that's good example. Synesthesia comes from cross coupling in the
brain of concepts that are usually separate. I synesthesia were like
then all synesthesists would see the same numbers as having the same color,
The main thing that causes us to attribute a form of existence to
objects is that everyone who understands them agrees on their properties.
> 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 numbers don't cause anything and they are not caused by other things. So
it's not an accident.
> 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.
It is more than just perceiving them differently. For example mathematical
objects are not located in space or time. They exist timelessly and in no
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