cksalmon wrote:I'm reading through John Polkinghhorne's
The God of Hope and the End of the World and came across this passage:
“Mathematics is the natural language of science.
…
Yet, what is mathematics itself? It’s practicioners resist the suggestion that it is a constructive form of intellectual play. They believe that their researches are true discoveries, explorations of an already-existing reality. The prime numbers and the Mandelbrot set have always been ‘there’. But ‘where’ have they been? If these convictions of the mathematicians are correct (and I believe them to be), then in addition to the physical world that the scientists investigate, there must be an everlasting noetic world of mathematical entities that the mathematicians investigate.”
While studying in seminary, I had a professor who was convinced that numbers actually “exist” in some sense. I was convinced, and perhaps still am, on the other hand, that numbers and the math that utilize them are mind-dependent constructs and that to believe that they have some sort of essence qua entities (in other words to ontologize them)—however fuzzily or creatively defined—was merely to engage in reifying abstractions.
In other words, my professor and Polkinghorne are platonists, at least with respect to numbers.
As I read the passage in
God of Hope, I started wondering how mathematicians view numbers. Do they “exist?” In what sense? Tarski? Asbestosman? Others?
Oh, Asbestosman is a Latter-day Saint. That’s this thread’s tenuous connection to Mormonism.
Thanks for your insight.
Chris
EDITED TO ADD: "math that utilize them..." should be "math..." The board is auto-correcting my post. Weird, but perhaps generally useful.
JAK:
May I add another dimension to your inquiry? Music is often regarded as the universal language.
Music is intimately connected with mathematics. Middle C is 256 cycles per second. That’s the fundamental frequency. However, a trumpet playing middle C sounds different than a clarinet playing the same note on the same fundamental frequency. It’s the
overtones which make the difference in sound to the human ear. But it
is mathematics. An orchestra of many players tunes to
A 440 cycles per second (mathematics). So music is
applied mathematics.
A pipe organ of more than 100 ranks (sets) of pipes has many orchestral voices. They are tuned in accordance with
mathematics given the individual rank of pipes. All
Principals are tuned to a fundamental frequency which matches all
string, all reeds, all flutes, etc. (I do not use “etc.” lightly). But, the organ is a simple and yet a complex instrument which is built from ground up based on mathematics – from the console (from which a single artist plays) to all the thousands of pipes of different lengths (different frequencies) are made.
Hence,
music is applied mathematics. Does
music exist? Most of us (especially those of us who play pipe organs) would argue “yes.” But more than affirmative, we (those who are musicians) would argue many other related elements relate to the art and artists who are masters as the console of a fine pipe organ. The organist plays
all the notes of the orchestra, not just one. As a result the artist (organist) has at his/her command the
mathematics of the instrument. No other single instrument in the world is parallel to that of the pipe organ. It’s history dates back many centuries, and it’s all an instrument constructed on
numbers.
May I share a few of many websites?:
Pipe Organ
World’s Largest Pipe Organs
Pipedreams
At the above website, many programs can be heard which feature the pipe organ (a mathematical invention). It relies on “numbers” and “mathematics” in its design and engineering.
Pipe organ example of a large pipe organ console which can be controlled by
single organist and which relies from the very first construction up to the many sets (ranks) of pipes to the entire instrument on
mathematical numbers. Every single pipe is tuned to a fundamental frequency (math).
You ask if these numbers are real or do they exist. For a company such as
Schantz, mathematics and numbers are
real as the organ factory builds organs shipped as far as
Melbourne, Australia
Such instruments are built on the premise that
mathematics is valid and reliable. At a cost of $4,217,417, the pipe organ built in Orrville, OH was constructed in that shop and installed as a tribute to
music first, but as a tribute to
mathematics as a reliable and fundamental basis of a musical instrument.
+++
And so your question:
“I started wondering how mathematicians view numbers. Do they “exist?” In what sense? Tarski? Asbestosman? Others?”
The answer is
yes in the world of music. But it is also
yes in the world of engineering of roads, bridges, buildings, etc.
I offer you but one example of
existence for “numbers.” It’s in an area rarely mentioned on this bb.
If you are in the least interested in the “king of instruments,” please Google additional sites which describe in detail the math involved in the engineering and construction of pipe organs. And that is only the beginning as the
artists which command these mathematical instruments are a marvel. Consider
Paul Jacobs who, at 31, has biography and performance which is
intimately linked to mathematics as an artist of our time.
JAK