Re: [math-fun] math, existence, and God
Forgive me if I'm repeating myself, but I feel inclined to reply to the one part of what Jim quoted that I understand -- a quote from an earlier post by Andy: << Back on March 25, Andy Latto wrote:
My view is that philosphers often worry too much about ontology (which things "exist" and which don't), when it really isn't a very interesting or important question, and it's just the nature of our language that misleads us into thinking it is.
Personally, I find existence (of certain things discussed in philosophy, anyway) to be of the utmost interest to me, much as is the case in mathematics. For example, does determinism exist? Or does perhaps the randomness that seems inherent to quantum mechanics preclude it? Do there exist other universes that are unable to interact with our own in any way? Is the apparent flow of time a real phenomenon -- something about time itself -- or it is only an artifact of our consciousness? Besides mathematical truth, there seems to exist a physical reality and an experiential reality (at least for each part of the universe, like a living person, capable of having experiences). Do these both truly exist? Are they merely two facets of the same thing? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On Tue, Jul 7, 2009 at 7:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Forgive me if I'm repeating myself, but I feel inclined to reply to the one part of what Jim quoted that I understand -- a quote from an earlier post by Andy:
<< Back on March 25, Andy Latto wrote:
My view is that philosphers often worry too much about ontology (which things "exist" and which don't), when it really isn't a very interesting or important question, and it's just the nature of our language that misleads us into thinking it is.
Personally, I find existence (of certain things discussed in philosophy, anyway) to be of the utmost interest to me, much as is the case in mathematics.
In mathematics, there are questions of existence that are interesting, and others that are uninteresting. Questions of what exists within a particular axiomatized mathematical system are often interesting: Does there exist an algorithm A that decides the traveling salesman problem in polynomial time? Does there exist an even number that is not the sum of two primes? Do there exist a, b, c, N, integers > 2, such that a^n + b^n = c^n are examples of the sort of existence question that mathematicians find interesting. When mathematicians first started using complex numbers, and found them useful for many purposes, but there was question as to whether they really existed. At some point, people realized that any statement about complex numbers is equivalent to a related statement about ordered pairs of reals. Assuming you're already convinced that the theory of the real numbers is without contradiction, this answers the interesting existence question, "Does there exist a proof of a contradiction using complex numbers?". But the question "Do complex numbers really exist, or is it only the ordered pairs of real numbers that exist" is an example of another sort of existence question, one that may be of interest to some philosophers of mathematics, but not usually to working mathematicians. The entities that exist in models of the ZF axiomatization of set theory are the sets. The entities that exist in models of the GB axiomatization include both the sets and the classes, where the classes are collections of sets that are "too large" to be sets, such as the class of all sets, and the class of all sets that are not members of themselves. Do classes "really exist"? Once it's been proved that the same statements about sets are stateable and provable in ZF and GB, and that the two theories of set theory are equiconsistent, I don't think that most mathematicians think the question "Yes, but do proper classes (that is, classes that are not sets) *really* exist to be a interesting one; many would not even consider it a meaningful one. My opinion is that philosophers worry to much about existence questions that are like the second sort of mathematical existence problem, and the appropriate answer is "who cares? there are equally good descriptions of the universe, with the same predictive power, that include and that fail to include those entities." Andy Latto andy.latto@pobox.com
On 7/8/09, Andy Latto <andy.latto@pobox.com> wrote:
... My opinion is that philosophers worry to much about existence questions that are like the second sort of mathematical existence problem, and the appropriate answer is "who cares? there are equally good descriptions of the universe, with the same predictive power, that include and that fail to include those entities."
To put it in a nutshell --- does existence exist? The mathematician is a magician --- with a wave of a stick of chalk, he incants a definition, and behold --- a new concept is summoned into existence. End of discussion. An empiricist (Dr. Johnson) must take a more sensorily committed line, and give the nearest boulder a good kick --- this sort of existence is implicit in the term "qualia" used earlier. But the existence of God, or of the (physical) quantum, or of various flavours of number --- Kronecker apocryphally observed that "God created the integers, the rest is the work of Man" --- is surely a question involving emotional and social factors. They are what Dawkins calls "memes" (what was wrong with "ideas", I wonder?) --- and as such, they plainly do exist, simply because enough people believe in them --- though this situation may well be temporary. In addition, at a sufficiently fine level of detail, individual perceptions of a meme will vary --- some smoothing process may become necessary to deliver an agreed (some hope!) "mean meme". Questions of empirical verifiability or internal consistency are irrelevant. Naive sets exist because Naive Set Theory exists, and that in turn because it is quite useful (even though it's wrong!). The aether exists because Newtonian mechanics exists, ditto. Fred Lunnon
On Wed, Jul 8, 2009 at 9:04 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Questions of empirical verifiability or internal consistency are irrelevant. Naive sets exist because Naive Set Theory exists, and that in turn because it is quite useful (even though it's wrong!). The aether exists because Newtonian mechanics exists, ditto.
I think the word existence is overloaded here. There is a sense in which phlogiston exists, because the theory that things are flammable because they contain phlogiston, which is released by burning, exists. There is also a sense in which phlogiston does not exist, and the statement "there's no such thing as phlogiston; when things burn, they are combining with oxygen, not releasing something; phlogiston (the thing which is released when something is burned) does not exist" is a true statement. I think that discussions of existence that don't make the difference between these two kinds of existence clear are not going to be useful. Andy
So, do holes exist? -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com]On Behalf Of Andy Latto Sent: Wednesday, July 08, 2009 09:33 To: math-fun Subject: Re: [math-fun] math, existence, and God On Wed, Jul 8, 2009 at 9:04 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Questions of empirical verifiability or internal consistency are irrelevant. Naive sets exist because Naive Set Theory exists, and that in turn because it is quite useful (even though it's wrong!). The aether exists because Newtonian mechanics exists, ditto.
I think the word existence is overloaded here. There is a sense in which phlogiston exists, because the theory that things are flammable because they contain phlogiston, which is released by burning, exists. There is also a sense in which phlogiston does not exist, and the statement "there's no such thing as phlogiston; when things burn, they are combining with oxygen, not releasing something; phlogiston (the thing which is released when something is burned) does not exist" is a true statement. I think that discussions of existence that don't make the difference between these two kinds of existence clear are not going to be useful. Andy _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Wholly.
So, do holes exist?
On Wed, Jul 8, 2009 at 8:15 AM, Mike Speciner <ms@alum.mit.edu> wrote:
So, do holes exist?
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com]On Behalf Of Andy Latto Sent: Wednesday, July 08, 2009 09:33 To: math-fun Subject: Re: [math-fun] math, existence, and God
On Wed, Jul 8, 2009 at 9:04 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Questions of empirical verifiability or internal consistency are irrelevant. Naive sets exist because Naive Set Theory exists, and that in turn because it is quite useful (even though it's wrong!). The aether exists because Newtonian mechanics exists, ditto.
I think the word existence is overloaded here. There is a sense in which phlogiston exists, because the theory that things are flammable because they contain phlogiston, which is released by burning, exists. There is also a sense in which phlogiston does not exist, and the statement "there's no such thing as phlogiston; when things burn, they are combining with oxygen, not releasing something; phlogiston (the thing which is released when something is burned) does not exist" is a true statement. I think that discussions of existence that don't make the difference between these two kinds of existence clear are not going to be useful.
Andy _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Thane Plambeck tplambeck@gmail.com http://thaneplambeck.typepad.com/
participants (5)
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Andy Latto -
Dan Asimov -
Fred lunnon -
Mike Speciner -
Thane Plambeck