[FOM] Convincing math-blind people that math is different
Franklin
franklin.vp at gmail.com
Thu Dec 25 04:56:25 EST 2014
On Wed, Dec 24, 2014 at 11:59 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> The demonstration is simple to describe. I build a computer and implement
> an algorithm that prints out, on paper, a million digits of some constant
> that hasn't been explicitly computed before---say, sqrt(12523599347). Then I
> build a completely different kind of computer and implement a completely
> different algorithm. I announce that my new system will print out exactly
> the same million digits. Then I hit the "go" button and the machine duly
> churns out the predicted million digits. The math-blind person can verify
> that the million digits are indeed the same.
>
> This sort of demonstration would seem to have no analogue in other fields of
> knowledge. For example, we might be able to find two people who are able to
> recite the entire Koran word-for-word, but this is because the Koran has
> already been written out explicitly for all to examine. The first million
> digits of sqrt(12523599347) have not been written out before, as far as the
> math-blind person can see. All non-scientific examples I can think of that
> involve agreed-upon conventions (e.g., laws, works of art) require that a
> community spend considerable time drawing up the conventions explicitly, and
> explicitly disseminating that knowledge. The way in which an algorithm
> encodes an enormous number of digits seems to be a uniquely
> mathematical/scientific phenomenon.
I cannot see the difference between this example and other physical
phenomena. I see the mathematical definition of sqrt(12523599347) as
a description of a physical experiment which consists on multiplying a
number by itself and obtaining 12523599347. I think that what is
challenging when comparing mathematical knowledge with other types of
knowledge is the simple nature of the former. Mathematical knowledge
tends to refer to physical experiments (occurring in the mind, on the
paper, or on a computer) that are highly idealized or abstracted. In
other areas of knowledge, experiments tend to involve more complicated
objects and the results tend to depend on many factors that may not be
feasible, or meaningful, to eliminate or to abstract. Add to this that
perhaps knowledge from other areas, if they have comparable
simplicity, could be called mathematical knowledge.
I think one experiment that seems similar in nature is to consider an
stable atom with a certain number of protons. The number of protons
here is serving the same role as the defining property of the square
root sqrt(12523599347) in the mathematical experiment. The stability
of the atom and the conditions in the laboratory are analogous to
conditions we assume in the mathematical experiment. Namely, we assume
the computations are carried by an agent that is able to follow the
intended algorithms. Now we can ask questions about the atom. For
example, questions about its atomic orbital. The properties of its
atomic orbital can be obtained through many different methods. These
serving as analogue to the different algorithms one can use to produce
digits of sqrt(12523599347). The properties of the atomic orbital
serving as analogue to the digits of sqrt(12523599347).
_____________________________
Franklin Vera Pacheco
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