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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 maths 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 "maths..." The board is auto-correcting my post. Weird, but perhaps generally useful.

Tarski is a neo-platonist.

I, on the other hand, tend to take your view that numbers are mind-dependent constructs.

I think that numbers are a natural abstraction of experience in the world. Also, If I recall correctly from philosophy, Kant may have had something to say on this--that geometry and arithmetic are abstractions of the mind's way of making sense of experiences throught concepts like time and space. I think it's this natural way of making sense of our world that causes us all to have similar perceptions about numbers.

I also think there may be some evidence of math in our universe and experience that could go either way--be either true or false. The Axiom of Choice is perhaps one such thing. I believe the continuum hypothesis may be another.

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

I remember Descartes pointing out that 2+2=4 in a dream as well as while awake. Even if all intelligent life were to go extinct, 2+2 would still equal 4.

To me that's a pretty strong argument for the existence of logic that is independent of the mind that recognizes it.

I guess it's like the old question, "If a tree falls in the woods and nobody is there to hear it, did it still make a sound." I would say, "Yes, it did make a sound." Does the sun still shine even if no eye sees it? Depending on how you choose to define sound or light, I suppose one answer is as valid as another.

asbestosman wrote:Tarski is a neo-platonist.

Well, kinda

I don't necessarily think numbers have an actual existence. It is just that some things are possible to think consistently by any intelligent being and some things are not. Therefore what is possible has a structure. But it is not actual until something instantiates that pattern.
It is similar to the library of mendel. Some genomes have survival value in a fixed environment and others do not. The set of all genomes has a structure but it need not be actualized by a real species.


The set of possible logical worlds seems structured to me--but that is not to say that they are actual worlds in some platonic heaven unless that heaven is just "that which is possible". It is just a question of where you want to put ontological weight.
The Mandelbrot set was always possible (or more precisely, reflects a possibility in computation of minds and machines) but not actual until it became instantiated in our thought or on a computer.
It isn't floating in an immaterial heaven in any spooky sense.

And PS, there is no proof of God in all this talk. None. Abstact possibilities have a strcutre but that can't represent a real being that acts in the world.

Tarski wrote:I don't necessarily think numbers have an actual existence. It is just that some things are possible to think consistently by any intelligent being and some things are not. Therefore what is possible has a structure. But it is not actual until something instantiates that pattern.
It is similar to the library of mendel. Some genomes have survival value in a fixed environment and others do not. The set of all genomes has a structure but it need not be actualized by a real species.


The set of possible logical worlds seems structured to me--but that is not to say that they are actual worlds in some platonic heaven unless that heaven is just "that which is possible". It is just a question of where you want to put ontological weight.
The Mandelbrot set was always possible (or more precisely, reflects a possibility in computation of minds and machines) but not actual until it became instantiated in our thought or on a computer.
It isn't floating in an immaterial heaven in any spooky sense.

I think that sort of resonates with me. But I'm not completely sure. Perhaps the Mandelbrot set isn't possible in computation of minds and machines in some possible worlds. The question that now comes up in my mind is about the set of possible worlds--is that set dependent on our point of view and our logic, or is there really some universal structure giving what is or is not possible. Is there a universal perspectibe for math and if so, how can we know what it is? Is there a universal perspective for all possible worlds? I can certainly imagine some math that isn't possible, or at least not consistent and therefore not useful so far as I can tell.

I also tend to think that perhaps some possible worlds are more probable than others. Perhaps set theory is very probable to be applicable in any possible world, or at least those where consciousness is possible.


I grant that there is a beauty in mathematics--a beauty I don't feel that I create myself. However, I'm not so sure what the "source" of that is. I often wonder if it isn't mostly just my own brain imprinting itself on my perceptions and thoughts and all human brains doing the same. Perhaps some aspects of math truly are universal, but I'm not convinced that this must be the case. But that's a philosophical question more than a mathematical one and I easiliy get myself confused with philosophy.

We are taught in Maths that Maths is just a language, but one likes to think of Maths as fundamental, because it helps in explaining things too complex for our minds to comprehend. I imagine that when we start to learn more, we will evelope more languages.

I am not a mathematician, even I'd like to be one.
I am an user.

My wife is (sorry, she was) a teacher of mathemathic and physics. I was a teacher of radar systems and many relative subject.
For her, math is something to teach. For me, math is something to use.

I use hammer, screwdriver, measuring devices, computers - and numbers, equations, formulas. Hammer, screwdriver, any electronic device are real things. For me, an equation is as real as a hammer. When I use them, the results are valid, tangible. Yes, there are categories. The objects of the hammer are more touchable (my finger, for example) than the objects of math.

Numbers are real things, if You want define differently, they are useful things. Unfortunately, many topic of these threads are not. They are irreal, not real, unreal - i am not certain if all these words are existing ones in english but I am certain You understand them. Should I make a short list of nonreals? Gospel, Holy Spirit, Devil, prophet (named or used in general sense), endowment, sacred, - STOP-ENOUGH - and last but not least, God/GOD/god/g_d (sometimes many of them).

Tarski wrote:And PS, there is no proof of God in all this talk. None. Abstact possibilities have a strcutre but that can't represent a real being that acts in the world.

ludwigm wrote:Should I make a short list of nonreals? Gospel, Holy Spirit, Devil, prophet (named or used in general sense), endowment, sacred, - STOP-ENOUGH - and last but not least, God/GOD/god/g_d (sometimes many of them).

Certainly, I have theistic concerns relative to this topic, but I'm not attempting to prove God with this thread. This is a philosophical conundrum for me rather than an apologetic endeavor.

I appreciate all the responses so far. I'm going to have to think some things through before I comment or ask anyone for clarification.

Best.

Chris

I don't know about the other numbers, but three leads a magical existence and zero is my hero.

Three is a Magic Number

Zero is My Hero!

"That's why with only ten numbers and zero, you can count to far as you can go--to infinity!

Forever and ever with zero, my hero, how wonderful you are!"


KA